23.31 - Miscellaneous Examples (3) on Integrals

In the previous section, we saw some miscellaneous examples on integrals. In this section, we will see a few more miscellaneous examples.

Solved Example 23.148
Integrate $\small{\frac{e^x}{(1+e^x)(2+e^x)}}$    
Solution:
1. Let us put $\small{u = 1+e^x}$

• Then $\small{\frac{du}{dx}~=~e^x \Rightarrow e^x\,dx~=~du}$

• Also, since $\small{u = 1+e^x}$, we can write:

$\small{2+e^x = u+1}$

2. So we want:

$\small{I = \int{\left[\frac{e^x}{(1+e^x)(2+e^x)}\right]dx}= \int{\left[\frac{1}{(u)(u+1)}\right]du}= \int{\left[\frac{1}{u^2 + u}\right]du}}$

3. This is a standard integral of the form:

$\small{ \int{\left[\frac{px + q}{ax^2 + bx + c}\right]du}}$, where p = 0, q = 1, a = 1, b = 1 and c = 0

We get: $\small{I~=~\int{\left[\frac{1}{u^2 + u}\right]du}~=~(-1)\log \left| \frac{1}{u}~+~1 \right|~+~\rm{C}}$

$\small{~=~(-1)\log \left| \frac{1+u}{u} \right|~+~\rm{C}~=~\log \left| \frac{u}{1+u} \right|~+~\rm{C}}$

4. Substituting for u, we get:

$\small{I~=~\log \left| \frac{1+e^x}{2+e^x} \right|~+~\rm{C}}$

Solved Example 23.149
Integrate $\small{f'(ax+b) \left[f(ax+b) \right]^n}$
Solution:
1. Let us put $\small{u = f(ax+b)}$

• Then $\small{\frac{du}{dx}~=~a\,f'(ax+b) \Rightarrow a\,f'(ax+b)\,dx~=~du}$

2. So we want:

$\small{I = \int{\left[f'(ax+b) \left[f(ax+b) \right]^n\right]dx}= \int{\left[\frac{(a)f'(ax+b) \left[f(ax+b) \right]^n}{a}\right]dx}}$

$\small{= \int{\left[\frac{u^n}{a}\right]du}}$

3. This integration gives: $\small{I = \frac{u^{n+1}}{a(n+1)}~+~\rm{C}}$

4. Substituting for u, we get:

$\small{I = \frac{[f(ax+b]^{n+1}}{a(n+1)}~+~\rm{C}}$

Solved Example 23.150
Integrate $\small{\left(\frac{2 + \sin(2x)}{1 + \cos(2x)} \right)e^x}$
Solution:
1. 1. First we will rearrange the given expression:

$\small{\left(\frac{2 + \sin(2x)}{1 + \cos(2x)} \right)e^x~=~\left(\frac{2 + 2\sin(x) \cos (x)}{2 \cos^2 x} \right)e^x}$

$\small{~=~\left(\frac{1}{\cos^2 x}~+~\tan x \right)e^x~=~\left(\sec^2 x~+~\tan x \right)e^x}$

2. So we want:

$\small{I = \int{\left[\left(\sec^2 x~+~\tan x \right)e^x\right]dx}}$

• $\small{\sec^2 x}$ is the derivative of $\small{\tan x}$. So this is of the form:

$\small{\int{\left[\left(f(x)~+~f'(x) \right)e^x\right]dx}~=~e^x\,f(x)~+~\rm{C}}$

3. Thus we get:

$\small{I = \int{\left[\left(\frac{2 + \sin(2x)}{1 + \cos(2x)} \right)e^x\right]dx}= \int{\left[\left(\sec^2 x~+~\tan x \right)e^x\right]dx}~=~e^x\,\tan(x)~+~\rm{C}}$

Solved Example 23.151
Integrate $\small{\frac{\sqrt{x^2 + 1}\left[\log(x^2 + 1)~-~2 \log x \right]}{x^4}}$
Solution:
1. First we will rearrange the given expression:

Put $\small{x~=~\tan u}$

$\small{\Rightarrow 1+x^2~=~1+\tan^2 u~=~\sec^2 u}$

• Also, $\small{\frac{dx}{du}~=~\sec^2 u \Rightarrow dx~=~\sec^2 u \, du}$

• So we can write:

$\small{I~=~\int{\left[\frac{\sqrt{x^2 + 1}\left[\log(x^2 + 1)~-~2 \log x \right]}{x^4} \right]dx}~=~\int{\left[\frac{\sqrt{\sec^2 u}\left[\log(\sec^2 u)~-~ \log (\tan^2 u) \right]}{\tan^4 u} \right]\sec^2 u\,du}}$

$\small{~=~\int{\left[\frac{\left[\log\left(\frac{\sec^2 u}{\tan^2 u} \right) \right]}{\tan^4 u} \right]\sec^3 u\,du}~=~\int{\left[\frac{\left[\log\left(\frac{1}{\sin^2 u} \right) \right]}{\tan^4 u\left(\cos^3 u \right)} \right]du}}$

$\small{~=~\int{\left[\frac{\cos u\left[\log\left(\frac{1}{\sin^2 u} \right) \right]}{\sin^4 u} \right]du}}$

2. We will use the method of integration by parts.

(a) Assigning first and second functions:

   ♦ Let first function be: $\small{f(u)=\log\left(\frac{1}{\sin^2 u} \right)}$

   ♦ Let second function be: $\small{g(u)=\frac{\cos u}{\sin^4 u}}$

(b) Finding A:

$\small{A~=~\int{\left[g(u) \right]dx}~=~\frac{- 1}{3\sin^3 u}}$

(c) $\small{\big[f(u) \left(A \right) \big]~=~\big[ \log\left(\frac{1}{\sin^2 u} \right)\big] \,\big[\frac{- 1}{3 \sin^3 u} \big]}$

• This is the first term.

(d) $\small{f'(u)~=~\frac{-2 \cos(u)}{\sin(u)}}$

(e) $\small{\int{\big[f'(u)\,\left(A \right)  \big]du}~=~\int{\big[\frac{-2 \cos(u)}{\sin(u)}\,\left(\frac{- 1}{3 \sin^3 (u)} \right)\big]du}}$

$\small{~=~\frac{2}{3}\int{\big[\frac{\cos(u)}{\sin^4(u)}\big]du}~=~\frac{-2}{9\sin^3 u}}$

• This is the second term.

(f) So we get:

$\small{I~=~\text{First term - Second term}}$

$\small{~=~\big[ \log\left(\frac{1}{\sin^2 u} \right)\big] \,\big[\frac{- 1}{3 \sin^3 u} \big]~-~\big[\frac{-2}{9 \sin^3 u} \big]}$

$\small{~=~\big[ \log\left(\frac{1}{\sin^2 u} \right)\big] \,\big[\frac{- 1}{3 \sin^3 u} \big]~+~\big[\frac{2}{9 \sin^3 u} \big]}$

$\small{~=~\big[\frac{2}{9 \sin^3 u} \big]~-~\big[ \log\left(\frac{1}{\sin^2 u} \right)\big] \,\big[\frac{1}{3 \sin^3 u} \big]}$

$\small{~=~\big[\frac{1}{3\sin^3 u} \big]\bigg[\frac{2}{3}~-~ \log\left(\frac{1}{\sin^2 u} \right)\bigg]}$

3. Now we will substitute for u:

(a) We wrote $\small{x~=~\tan u}$

$\small{\Rightarrow 1+x^2~=~1+\tan^2 u~=~\sec^2 u}$

$\small{\Rightarrow \cos^2 u~=~\frac{1}{1+x^2}}$

$\small{\Rightarrow \sin^2 u~=~1~-~\frac{1}{1+x^2}~=~\frac{x^2}{1+x^2}}$

$\small{\Rightarrow \frac{1}{\sin^2 u}~=~1~+~\frac{1}{x^2}}$

$\small{\Rightarrow \frac{1}{\sin u}~=~\left(1~+~\frac{1}{x^2} \right)^{1/2}}$

$\small{\Rightarrow \frac{1}{\sin^3 u}~=~\left(1~+~\frac{1}{x^2} \right)^{3/2}}$

(b) So we get:

$\small{I~=~\big[\frac{1}{3\sin^3 u} \big]\bigg[\frac{2}{3}~-~ \log\left(\frac{1}{\sin^2 u} \right)\bigg]}$

$\small{~=~\big[\frac{1}{3} \left(1~+~\frac{1}{x^2} \right)^{3/2} \big]\bigg[\frac{2}{3}~-~ \log\left(1~+~\frac{1}{x^2} \right)\bigg]~+~\rm{C}}$

Solved Example 23.152
Integrate $\small{\frac{1}{e^x~+~e^{-x}}}$
Solution:
1. First we will rearrange the given expression:

$\small{\frac{1}{e^x~+~e^{-x}}~=~\frac{1}{e^x~+~\frac{1}{e^x}}~=~\frac{e^x}{e^{2x}~+~1}}$

Put $\small{u~=~e^x}$

• Then, $\small{\frac{du}{dx}~=~e^x \Rightarrow e^x\,dx ~=~du}$

2. So we can write:

$\small{I~=~\int{\left[\frac{1}{e^x~+~e^{-x}} \right]dx}~=~\int{\left[\frac{e^x}{e^{2x}~+~1} \right]du}~=~\int{\left[\frac{1}{u^{2}~+~1} \right]du}}$

3. This is a standard integral. We get:

$\small{I~=~\tan^{-1}u~+~\rm{C}}$

4. Substituting for u, we get:

$\small{I~=~\tan^{-1}(e^x)~+~\rm{C}}$

Solved Example 23.153
Evaluate $\small{\int_0^1{\left[e^{2 - 3x} \right]dx}}$ as a limit of a sum.
Solution:
1. 1. $\small{\int_{0}^{1}{\left[e^{2-3x} \right]dx}}$ is the area bounded by the four items:
   ♦ The curve $\small{y = f(x) = e^{2-3x}}$
   ♦ The vertical line x = 0 (y-axis)
   ♦ The vertical line x = 1
   ♦ The horizontal line y = 0 (x-axis)
   
2. We have:
$\small{\int_a^b{\left[f(x) \right]dx}~=~\lim_{n\rightarrow \infty} \frac{b-a}{n} \left[f(a+h)~+~f(a+2h)~+~.~.~.~+~f(a+nh) \right]}$

• In our present case, a = 0 and b = 1

So $\small{h~=~\frac{b-a}{n}~=~\frac{1-0}{n}~=~\frac{1}{n}}$

3. Now the formula becomes:
Area =
$\small{\lim_{n\rightarrow \infty} \frac{b-a}{n} \left[f(a+h)~+~f(a+2h)~+~f(a+3h)~+~~.~.~.~+f(a+nh) \right]}$

$\small{~=~\lim_{n\rightarrow \infty} \frac{1}{n} \left[f(a+\frac{1}{n})~+~f(a+\frac{2}{n})~+~f(a+\frac{3}{n})~+~~.~.~.~+f(a+(n)\frac{1}{n}) \right]}$

$\small{~=~\lim_{n\rightarrow \infty} \frac{1}{n} \bigg[\big[e^{2-3(a+\frac{1}{n})}\big]~+~\big[e^{2-3(a+\frac{2}{n})}\big]~+~\big[e^{2-3(a+\frac{3}{n})}\big]~+~~.~.~.~}$

$\small{~~~~~~~~~.~.~.~+~\big[e^{2-3(a+(n)\frac{1}{n})}\big]\bigg]}$

4. Let us determine the quantity inside the large square brackets. We have to do a summation:

$\small{e^{2-3(a+\frac{1}{n})}~+~e^{2-3(a+\frac{2}{n})}~+~e^{2-3(a+\frac{3}{n})}~+~.~.~.~ \text{n terms}}$

$\small{~=~\left(e^2~\times~e^{-3a}~\times~e^{\frac{-3}{n}} \right)~+~\left(e^2~\times~e^{-3a}~\times~e^{\frac{-6}{n}} \right)~+~\left(e^2~\times~e^{-3a}~\times~e^{\frac{-9}{n}} \right)~+~.~.~.~ \text{n terms}}$

$\small{~=~e^{2-3a}\left(e^{\frac{-3}{n}}~+~e^{\frac{-6}{n}}~+~e^{\frac{-9}{n}}~+~.~.~.~ \text{n terms} \right)}$

$\small{~=~e^{2}\left(e^{\frac{-3}{n}}~+~e^{\frac{-6}{n}}~+~e^{\frac{-9}{n}}~+~.~.~.~ \text{n terms} \right)}$

$\small{~~~~~~\because {3a}~=~{3(0)}~=~0}$

• So inside the brackets, we have a geometric progression.

• First term = $\small{e^{\frac{-3}{n}}}$

• Common ratio  $\small{~r~=~\frac{e^{\frac{-6}{n}}}{e^{\frac{-3}{n}}}~=~\frac{e^{\frac{-9}{n}}}{e^{\frac{-6}{n}}}~=~e^{\frac{-3}{n}}}$

• Sum of the first n terms is given by: $\small{\frac{\text{First term}~\times~\left(r^{n-1}~-~1 \right)}{r~-~1}}$

$\small{r^{n-1}~=~\left(e^{\frac{-3}{n}}\right)^{n-1}~=~e^{\frac{-3n+3}{n}}~=~e^{-3+\frac{3}{n}}}$

• So we get:

Sum of all terms of the G.P

$\small{~=~\frac{e^{\frac{-3}{n}}~\times~\left({e^{-3+\frac{3}{n}}}~-~1 \right)}{e^{\frac{-3}{n}}~-~1}~=~\frac{{e^{-3}}~-~e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1}}$

• So the summation is:

$\small{e^2 \left[\frac{{e^{-3}}~-~e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right]}$

5. So the limit in (3) becomes:

$\small{\lim_{n\rightarrow \infty} \frac{1}{n} \Bigg[e^2 \left[\frac{{e^{-3}}~-~e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right] \Bigg]}$

$\small{~=~e^2 \lim_{n\rightarrow \infty} \frac{1}{n}  \left[\frac{{e^{-3}}~-~e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right]}$

$\small{~=~e^2 \lim_{n\rightarrow \infty} \frac{1}{n}  \left[\frac{{e^{-3}} }{e^{\frac{-3}{n}}~-~1} \right]~-~e^2 \lim_{n\rightarrow \infty} \frac{1}{n}  \left[\frac{e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right]}$

$\small{~=~e^2 \lim_{n\rightarrow \infty} \frac{(-3)}{(-3)n}  \left[\frac{{e^{-3}} }{e^{\frac{-3}{n}}~-~1} \right]~-~e^2 \lim_{n\rightarrow \infty} \frac{(-3)}{(-3)n}  \left[\frac{e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right]}$

$\small{~=~\frac{e^2}{(-3)} \lim_{n\rightarrow \infty} \frac{-3}{n}  \left[\frac{{e^{-3}} }{e^{\frac{-3}{n}}~-~1} \right]~-~\frac{e^2}{(-3)} \lim_{n\rightarrow \infty} \frac{-3}{n}  \left[\frac{e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right]}$

$\small{~=~\frac{e^{-1}}{(-3)} \lim_{n\rightarrow \infty} \frac{-3}{n}  \left[\frac{{1} }{e^{\frac{-3}{n}}~-~1} \right]~-~\frac{e^2}{(-3)} \lim_{n\rightarrow \infty} \frac{-3}{n}  \left[\frac{e^{\frac{-3}{n}} }{e^{\frac{-3}{n}}~-~1} \right]}$

6. In the above expression, the limits can be evaluated as follows:

$\small{\frac{e^{-1}}{(-3)} \lim_{n\rightarrow \infty}   \left[\frac{{1} }{\frac{e^{\frac{-3}{n}}~-~1}{\frac{-3}{n}}} \right]
~-~\frac{e^2}{(-3)} \lim_{n\rightarrow \infty}   \left[\frac{{e^{\frac{-3}{n}}} }{\frac{e^{\frac{-3}{n}}~-~1}{\frac{-3}{n}}}  \right]}$

$\small{~=~\frac{e^{-1}}{(-3)}  \bigg[\frac{1}{1} \bigg]~-~\frac{e^2}{(-3)}\big[\frac{1}{1} \big]~=~\frac{e^{-1}~-~e^2}{(-3)}}$

$\small{~=~\frac{e^{2}~-~e^{-1}}{3}}$

• Here we use two facts:

(i) $\small{\lim_{n\rightarrow \infty}  \left[e^{\frac{-3}{n}} \right]~=~e^{\frac{-3}{\infty}}~=~e^0~=~1}$

(ii) Let $\small{\frac{-3}{n}~=~h}$.
Then $\small{h \rightarrow 0 ~\text{as}~n \rightarrow \infty}$

So $\small{\lim_{n\rightarrow \infty}  \Big[\frac{e^{\frac{-3}{n}}~-~1}{\frac{-3}{n}}\Big]~=~\lim_{n\rightarrow \infty}  \Big[\frac{e^{h}~-~1}{h}\Big]~=~1}$   

Solved Example 23.154
Evaluate $\small{\int_{0}^{\frac{\pi}{4}}{\left[\frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \right]dx}}$
Solution:
1. Let us rearrange the given expression:

$\small{\frac{\sin x \cos x}{\cos^4 x + \sin^4 x}~=~\frac{\sin x \cos x}{\cos^4 x + \sin^4 x~+~2\sin^2x\,\cos^2x~-~2\sin^2x\,\cos^2x}}$

$\small{~=~\frac{\sin x \cos x}{\left(\cos^2 x + \sin^2 x \right)^2~-~2\sin^2x\,\cos^2x}~=~\frac{\sin x \cos x}{1~-~2\sin^2x\,\cos^2x}}$

$\small{~=~\frac{\sin x \cos x}{1~-~(4\sin^2x\,\cos^2x)/2}~=~\frac{2\sin x \cos x}{2~-~4\sin^2x\,\cos^2x}}$

$\small{~=~\frac{\sin (2x)}{2~-~\sin^2(2x)}~=~\frac{\sin (2x)}{1+[1~-~\sin^2(2x)]}~=~\frac{\sin (2x)}{1+[\cos^2(2x)]}}$

2. So we want: $\small{I~=~\int_{0}^{\frac{\pi}{4}}{\left[\frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \right]dx}~=~\int_{0}^{\frac{\pi}{4}}{\left[\frac{\sin (2x)}{1+\cos^2(2x)} \right]dx}}$

3. First we will find the indefinite integral:

$\small{F~=~\int{\left[\frac{\sin (2x)}{1+\cos^2(2x)} \right]dx}}$

• Put $\small{u = \cos(2x)}$

Then $\small{\frac{du}{dx}~=~-2\sin(2x) \Rightarrow -2\sin(2x)\,dx~=~du}$

• We want:

$\small{F~=~\int{\left[\frac{(-2)\sin (2x)}{(-2)\left[1+\cos^2(2x) \right]} \right]dx}~=~\int{\left[\frac{1}{(-2)\left[1+u^2 \right]} \right]dx}~=~\frac{1}{(-2)}\int{\left[\frac{1}{1+u^2 } \right]dx}}$

• This is a standard integral. We get:

$\small{F~=~\left(\frac{-1}{2} \right)\tan^{-1}u}$

• Subsituting for u, we get:

$\small{F~=~\left(\frac{-1}{2} \right)\tan^{-1}\left[\cos(2x) \right]}$

4. Now we can evaluate the definite integral. We get:

$\small{I~=~\int_{0}^{\frac{\pi}{4}}{\left[\frac{\sin (2x)}{1+\cos^2(2x)} \right]dx}}$

$\small{~=~\big[\left(\frac{-1}{2} \right)\tan^{-1}\left[\cos(2x) \right]\big]_{0}^{\frac{\pi}{4}}}$

$\small{~=~\left(\frac{-1}{2} \right) \big[\tan^{-1}\left[\cos(2x) \right]\big]_{0}^{\frac{\pi}{4}}}$

$\small{~=~\left(\frac{-1}{2} \right) \big[\tan^{-1}\left[\cos\left(\frac{\pi}{2} \right) \right]~-~\tan^{-1}\left[\cos\left(0 \right) \right]\big]}$

$\small{~=~\left(\frac{-1}{2} \right) \big[\tan^{-1}\left[0 \right]~-~\tan^{-1}\left[1 \right]\big]}$

$\small{~=~\left(\frac{-1}{2} \right) \big[0~-~\frac{\pi}{4}\big]~=~\frac{\pi}{8}}$

Solved Example 23.155
Evaluate $\small{\int_{\frac{\pi}{2}}^{\pi}{\left[e^x\left(\frac{1-\sin x}{1 - \cos x} \right) \right]dx}}$
Solution:
1. Let us rearrange the portion inside braces:

$\small{\frac{1-\sin x}{1 - \cos x}~=~\frac{1-\sin x}{2 \sin^2\left(\frac{x}{2} \right)}~=~\frac{1}{2 \sin^2\left(\frac{x}{2} \right)}~-~\frac{\sin x}{2 \sin^2\left(\frac{x}{2} \right)}}$

$\small{~=~\frac{1}{2 \sin^2\left(\frac{x}{2} \right)}~-~\frac{2 \sin\left(\frac{x}{2} \right) \cos\left(\frac{x}{2} \right)}{2 \sin^2\left(\frac{x}{2} \right)}}$

$\small{~=~\frac{\csc^2\left(\frac{x}{2} \right)}{2}~-~\cot\left(\frac{x}{2} \right)~=~(-1)\left[\cot\left(\frac{x}{2} \right)~-~\frac{\csc^2\left(\frac{x}{2} \right)}{2} \right]}$

$\small{~=~(-1)\left[\cot\left(\frac{x}{2} \right)~+~\frac{(-1)\csc^2\left(\frac{x}{2} \right)}{2} \right]}$

2. So the given expression can be written as:

$\small{(-1)\,e^x\left[\cot\left(\frac{x}{2} \right)~+~\frac{(-1)\csc^2\left(\frac{x}{2} \right)}{2} \right]}$

• In the above result, $\small{\left[\frac{(-1)\csc^2\left(\frac{x}{2} \right)}{2} \right]}$ is the derivative of $\small{\left[\cot\left(\frac{x}{2} \right) \right]}$

• So the given expression can be written as:

$\small{(-1)\,e^x\left[f(x)~+~f'(x) \right]}$

3. So the indefinite integral can be written as:

$\small{F~=~(-1)\,e^x\left[f(x)\right]~=~(-1)\,e^x\left[\cot\left(\frac{x}{2} \right)\right]}$

4. Now we can evaluate the definite integral. We get:

$\small{I~=~\int_{\frac{\pi}{2}}^{\pi}{\left[e^x\left(\frac{1-\sin x}{1 - \cos x} \right) \right]dx}}$

$\small{~=~\big[(-1)\,e^x\left[\cot\left(\frac{x}{2} \right)\right]\big]_{\frac{\pi}{2}}^{\pi}}$

$\small{~=~\big[(-1)\,e^{\pi}\left[\cot\left(\frac{\pi}{2} \right)\right]\big]~-~\big[(-1)\,e^{\frac{\pi}{2}}\left[\cot\left(\frac{\pi}{4} \right)\right]\big]}$

$\small{~=~\big[(-1)\,e^{\pi}\left[0\right]\big]~-~\big[(-1)\,e^{\frac{\pi}{2}}\left[1\right]\big]}$

$\small{~=~e^{\frac{\pi}{2}}}$

Solved Example 23.156
Evaluate $\small{\int_{0}^{\frac{\pi}{2}}{\left[\frac{\cos^2 x}{\cos^2 x ~+~ 4 \sin^2 x} \right]dx}}$
Solution:
1. Let us rearrange the given expression:

$\small{\frac{\cos^2 x}{\cos^2 x ~+~ 4 \sin^2 x}~=~\frac{\cos^2 x}{\cos^2 x ~+~ 4(1-\cos^2 x)}~=~\frac{\cos^2 x}{4~-~3 \cos^2 x}}$

$\small{~=~\left(\frac{-1}{3} \right)\left[\frac{(-3)\cos^2 x}{4~-~3 \cos^2 x} \right]~=~\left(\frac{-1}{3} \right)\left[\frac{4~-~3\cos^2 x~-~4}{4~-~3 \cos^2 x} \right]}$

$\small{~=~\left(\frac{-1}{3} \right)\left[1~-~\frac{4}{4~-~3 \cos^2 x} \right]~=~\left(\frac{-1}{3} \right)\left[1~-~\frac{\frac{4}{\cos^2 x}}{\frac{4}{\cos^2 x}~-~\frac{3\cos^2 x}{\cos^2 x}} \right]}$

$\small{~=~\left(\frac{-1}{3} \right)\left[1~-~\frac{4 \sec^2 x}{4 \sec^2 x~-~3} \right]~=~\left(\frac{-1}{3} \right)\left[1~-~\frac{4 \sec^2 x}{4 (1+\tan^2 x)~-~3} \right]}$

$\small{~=~\left(\frac{-1}{3} \right)\left[1~-~\frac{4 \sec^2 x}{1~+~4 \tan^2 x} \right]}$ 

2. Now we can find the indefinite integral:

$\small{F~=~\int{\left[\frac{\cos^2 x}{\cos^2 x ~+~ 4 \sin^2 x} \right]dx}~=~\int{\bigg[\left(\frac{-1}{3} \right)\left[1~-~\frac{4 \sec^2 x}{1~+~4 \tan^2 x} \right] \bigg]dx}}$


$\small{~=~\int{\bigg[\left(\frac{-1}{3} \right) \bigg]dx}~+~\left(\frac{1}{3} \right)\int{\bigg[\frac{4 \sec^2 x}{1~+~4 \tan^2 x}  \bigg]dx}}$

$\small{~=~\frac{-x}{3}~+~\left(\frac{1}{3} \right)\int{\bigg[\frac{4 \sec^2 x}{1~+~4 \tan^2 x}  \bigg]dx}}$

$\small{~=~F_1~+~F_2}$

• $\small{F_2}$ can be calculated as follows:

• Put $\small{u~=~\tan x}$. Then $\small{\frac{du}{dx}~=~\sec^2 x \Rightarrow \sec^2 x\,dx~=~du}$

• So we get:

$\small{F_2~=~\left(\frac{1}{3} \right)\int{\bigg[\frac{4}{1~+~4(u)^2}  \bigg]dx}~=~\left(\frac{4}{3} \right)\int{\bigg[\frac{1}{1~+~(2u)^2}  \bigg]dx}}$

$\small{~=~\left(\frac{4}{3} \right)\left(\frac{1}{2} \right)\bigg[\tan^{-1}(2u)  \bigg]~=~\left(\frac{2}{3} \right)\bigg[\tan^{-1}(2 \tan x)  \bigg]}$

• Therefore, $\small{F~=~\frac{-x}{3}~+~\left(\frac{2}{3} \right)\left[\tan^{-1}(2 \tan x)  \right]}$

3. Now we can evaluate the definite integral. We get:

$\small{I~=~\big[\frac{-x}{3}~+~\left(\frac{2}{3} \right)\left[\tan^{-1}(2 \tan x)  \right]\big]_{0}^{\frac{\pi}{2}}}$

$\small{~=~\big[\frac{-\pi}{6}~+~\left(\frac{2}{3} \right)\left[\tan^{-1}(2 \tan \frac{\pi}{2})  \right]\big]~-~\big[0~+~\left(\frac{2}{3} \right)\left[\tan^{-1}(2 \tan 0)  \right]\big]}$

$\small{~=~\big[\frac{-\pi}{6}~+~\left(\frac{2}{3} \right)\left[\tan^{-1}(\infty)  \right]\big]~-~\big[0~+~\left(\frac{2}{3} \right)\left[\tan^{-1}(0)  \right]\big]}$

$\small{~=~\big[\frac{-\pi}{6}~+~\left(\frac{2}{3} \right)\left[\frac{\pi}{2}  \right]\big]~-~\big[0~+~\left(\frac{2}{3} \right)\left[0  \right]\big]}$

$\small{~=~\big[\frac{-\pi}{6}~+~\frac{2\pi}{6}\big]~=~\frac{\pi}{6}}$

Solved Example 23.157
Evaluate $\small{\int_{0}^{1}{\left[\frac{1}{\sqrt{1+x}~-~\sqrt{x}} \right]dx}}$
Solution:
1. Let us rearrange the given expression:

$\small{\frac{1}{\sqrt{1+x}~-~\sqrt{x}}~=~\frac{1(\sqrt{1+x}~+~\sqrt{x})}{(\sqrt{1+x}~-~\sqrt{x})(\sqrt{1+x}~+~\sqrt{x})}}$

$\small{~=~\frac{\sqrt{1+x}~+~\sqrt{x}}{1+x~-~x}~=~\sqrt{1+x}~+~\sqrt{x}}$

2. So we can write:

$\small{\int_{0}^{1}{\left[\frac{1}{\sqrt{1+x}~-~\sqrt{x}} \right]dx}~=~\int_{0}^{1}{\left[\sqrt{1+x}~+~\sqrt{x} \right]dx}}$

3. Now we can write the indefinite integral:

$\small{F~=~\int{\left[\sqrt{1+x}~+~\sqrt{x} \right]dx}~=~\frac{(1+x)^{3/2}}{3/2}~+~\frac{(x)^{3/2}}{3/2}}$

$\small{~=~\frac{(1+x)^{3/2}~+~(x)^{3/2}}{3/2}}$

4. Now we can evaluate the definite integral. We get:

$\small{I~=~\int_{0}^{1}{\left[\sqrt{1+x}~+~\sqrt{x} \right]dx}}$

$\small{~=~\big[\frac{(1+x)^{3/2}~+~(x)^{3/2}}{3/2}\big]_{0}^{1}}$

$\small{~=~\big[\frac{(1+1)^{3/2}~+~(1)^{3/2}}{3/2}\big]~-~\big[\frac{(1+0)^{3/2}~+~(0)^{3/2}}{3/2}\big]}$

$\small{~=~\big[\frac{2^{3/2}~+~1}{3/2}\big]~-~\big[\frac{1~+~0}{3/2}\big]~=~\frac{2^{3/2}}{3/2}~=~\frac{2^{5/2}}{3}~=~\frac{4 \sqrt2}{3}}$

Solved Example 23.158
Evaluate $\small{\int_{0}^{\pi}{\left[\frac{x \tan x}{\sec x~+~\tan x} \right]dx}}$
Solution:
1. Let us rearrange the given expression:

$\small{\frac{x \tan x}{\sec x~+~\tan x}~=~\frac{x \frac{\sin x}{\cos x}}{\frac{1}{\cos x}~+~\frac{\sin x}{\cos x}}~=~\frac{x \sin x}{1~+~\sin x}}$

2. So we want:

$\small{I~=~\int_{0}^{\pi}{\left[\frac{x \tan x}{\sec x~+~\tan x} \right]dx}~=~\int_{0}^{\pi}{\left[\frac{x \sin x}{1~+~\sin x} \right]dx}}$

3. Applying P₄, we get:

$\small{I~=~\int_{0}^{\pi}{\left[\frac{(\pi-x) \sin (\pi-x)}{1~+~\sin (\pi-x)} \right]dx}~=~\int_{0}^{\pi}{\left[\frac{(\pi-x) \sin x}{1~+~\sin x} \right]dx}}$

$\small{~=~\int_{0}^{\pi}{\left[\frac{\pi \sin x}{1~+~\sin x} \right]dx}~-~\int_{0}^{\pi}{\left[\frac{ x \sin x}{1~+~\sin x} \right]dx}}$

$\small{\Rightarrow I~=~\int_{0}^{\pi}{\left[\frac{\pi \sin x}{1~+~\sin x} \right]dx}~-~I}$

$\small{\Rightarrow 2I~=~\int_{0}^{\pi}{\left[\frac{\pi \sin x}{1~+~\sin x} \right]dx}}$

$\small{\Rightarrow I~=~\frac{\pi}{2}\int_{0}^{\pi}{\left[\frac{ \sin x}{1~+~\sin x} \right]dx}}$

$\small{\Rightarrow I~=~\frac{\pi}{2}\int_{0}^{\pi}{\left[\frac{\sin x (1~-~\sin x)}{(1~+~\sin x)(1~-~\sin x)} \right]dx}}$

$\small{~=~\frac{\pi}{2} \int_{0}^{\pi}{\left[\frac{\sin x~-~\sin^2 x}{1~-~\sin^2 x} \right]dx}~=~\frac{\pi}{2} \int_{0}^{\pi}{\left[\frac{\sin x~-~\sin^2 x}{\cos^2 x} \right]dx}}$

$\small{~=~\frac{\pi}{2} \int_{0}^{\pi}{\left[\sec x\, \tan x~-~\tan^2 x \right]dx}}$

$\small{~=~\frac{\pi}{2} \int_{0}^{\pi}{\left[\sec x\, \tan x~-~\left(\sec^2 x - 1 \right) \right]dx}}$

$\small{\Rightarrow I~=~\frac{\pi}{2} \int_{0}^{\pi}{\left[\sec x\, \tan x \right]dx}~-~\frac{\pi}{2} \int_{0}^{\pi}{\left[\sec^2 x \right]dx}~+~\frac{\pi}{2} \int_{0}^{\pi}{\left[1 \right]dx}}$

$\small{\Rightarrow I~=~I_1~-~I_2~+~I_3}$

4. First we will calculate I₁:

$\small{F_1~=~\frac{\pi}{2} \int{\left[\sec x\, \tan x \right]dx}~=~\frac{\pi}{2}\,\sec x}$

$\small{\Rightarrow I_1~=~\left[\frac{\pi}{2}\,\sec x \right]_{0}^{\pi}~=~\left[\frac{\pi}{2}\,\sec \left(\pi \right) \right]~-~\left[\frac{\pi}{2}\,\sec \left(0 \right) \right]}$

$\small{\Rightarrow I_1~=~\left[\frac{-\pi}{2} \right]~-~\left[\frac{\pi}{2} \right]~=~-\pi}$

5. Next we will calculate I₂:

$\small{F_2~=~\frac{\pi}{2} \int{\left[\sec^2 x \right]dx}~=~\frac{\pi}{2}\,\tan x}$

$\small{\Rightarrow I_2~=~\left[\frac{\pi}{2}\,\tan x \right]_{0}^{\pi}~=~\left[\frac{\pi}{2}\,\tan \left(\pi \right) \right]~-~\left[\frac{\pi}{2}\,\tan \left(0 \right) \right]}$

$\small{\Rightarrow I_2~=~\left[0 \right]~-~\left[0 \right]~=~0}$

6. Finally we will calculate I₃:

$\small{F_3~=~\frac{\pi}{2} \int{\left[1\right]dx}~=~\frac{\pi}{2}\,x}$

$\small{\Rightarrow I_3~=~\left[\frac{\pi}{2}\,(x) \right]_{0}^{\pi}~=~\left[\frac{\pi}{2}\, \left(\pi \right) \right]~-~\left[\frac{\pi}{2} \left(0 \right) \right]~=~\frac{\pi^2}{2}}$

7. From (3), (4), (5) and (6), we get:

$\small{I~=~I_1~-~I_2~+~I_3}$

$\small{~=~-\pi~-~0~+~\frac{\pi^2}{2}}$

$\small{~=~\left(\frac{\pi^2}{2}~-~\pi \right)~=~\left(\frac{\pi^2}{2}~-~\frac{2 \pi}{2} \right)}$

$\small{~=~\frac{\pi}{2}\left(\pi~-~2 \right)}$

Solved Example 23.159
Evaluate $\small{\int_{0}^{\frac{\pi}{4}}{\left[\frac{\sin x~+~\cos x}{9~+~16 \sin(2x)} \right]dx}}$
Solution:
1. Put $\small{u = \sin x ~-~\cos x}$

• Then $\small{\frac{du}{dx}~=~\cos x ~+~\sin x}$

$\small{\Rightarrow dx(\cos x ~+~\sin x)~=~du}$

• Also, $\small{u^2~=~\sin^2 x~+~\cos^2 x~-~2 \sin x \cos x~=~1~-~2 \sin x \cos x}$

$\small{\Rightarrow u^2~=~1~-~\sin(2x)}$

$\small{\Rightarrow \sin(2x)~=~1~-~u^2}$

2. So we want:

$\small{F ~=~\int{\left[\frac{\sin x~+~\cos x}{9~+~16 \sin(2x)} \right]dx}~=~\int{\left[\frac{1}{9~+~16(1-u^2)} \right]du}}$

$\small{~=~\int{\left[\frac{1}{9~+~16-16u^2} \right]du}~=~\int{\left[\frac{1}{25-16u^2} \right]du}~=~\int{\left[\frac{1}{5^2-(4u)^2} \right]du}}$

3. This is a standard integral. We have:

$\small{\int{\left[\frac{1}{a^2~-~t^2} \right]dt}~=~\frac{1}{2a} \log \left| \frac{a+t}{a-t}  \right|~+~\rm{C}}$

In our present case, a = 5 and t = 4u

• So we get:

$\small{F~=~\int{\left[\frac{1}{5^2-(4u)^2} \right]du}~=~\frac{1}{2(5)(4)} \log \left| \frac{5+4u}{5-4u}  \right|~=~\frac{1}{40} \log \left| \frac{5+4u}{5-4u}  \right|}$

4. Substituting for u, we get:

$\small{F~=~\frac{1}{40} \log \left|\frac{5~+~4 \sin x~-~4 \cos x}{5~-~4 \sin x~+~4 \cos x}  \right|}$

5. Now we can find the definite integral:

$\small{I~=~\left[\frac{1}{40} \log \left|\frac{5~+~4 \sin x~-~4 \cos x}{5~-~4 \sin x~+~4 \cos x}  \right| \right]_0^{\frac{\pi}{4}}}$

$\small{~=~\left[\frac{1}{40} \log \left|\frac{5~+~4 \sin \left(\frac{\pi}{4} \right)~-~4 \cos \left(\frac{\pi}{4} \right)}{5~-~4 \sin \left(\frac{\pi}{4} \right)~+~4 \cos \left(\frac{\pi}{4} \right)}  \right| \right]~-~\left[\frac{1}{40} \log \left|\frac{5~+~4 \sin \left(0 \right)~-~4 \cos \left(0 \right)}{5~-~4 \sin \left(0 \right)~+~4 \cos \left(0 \right)}  \right| \right]}$

$\small{~=~\left[\frac{1}{40} \log \left|\frac{5~+~0}{5~-~0}  \right| \right]~-~\left[\frac{1}{40} \log \left|\frac{5~+~0~-~4}{5~-~0~+~4}  \right| \right]}$

$\small{~=~\left[\frac{1}{40} \log \left|1  \right| \right]~-~\left[\frac{1}{40} \log \left|\frac{1}{9}  \right| \right]}$

$\small{~=~\left[\frac{1}{40} (0) \right]~-~\left[\frac{1}{40} \log \left|\frac{1}{9}  \right| \right]~=~~-~\left[\frac{1}{40} \log \left|\frac{1}{9}  \right| \right]}$

$\small{~=~\frac{1}{40} \log \left(9 \right)}$

Solved Example 23.160
Evaluate $\small{\int_{0}^{\frac{\pi}{2}}{\left[\sin(2x) \tan^{-1}(\sin x) \right]dx}}$
Solution:
1. Put $\small{u = \sin x}$

• Then $\small{\frac{du}{dx}~=~\cos x \Rightarrow dx~=~\frac{du}{\cos x}}$

• We wrote: $\small{u = \sin x}$

    ♦ When x approaches 0, u approaches 0
    ♦ When x approaches $\small{\frac{\pi}{2}}$, u approaches 1

2. We want:

$\small{I = \int_{0}^{\frac{\pi}{2}}{\left[\sin(2x) \tan^{-1}(\sin x) \right]dx}}$

$\small{= \int_{0}^{\frac{\pi}{2}}{\left[2 \sin(x)\cos(x) \tan^{-1}(\sin x) \right]dx}}$

$\small{= \int_{0}^{\frac{\pi}{2}}{\left[2 u\cos(x) \tan^{-1}(u) \right]\frac{du}{\cos x}}}$

$\small{= \int_{0}^{1}{\left[2 u \tan^{-1}(u) \right]du}= 2 \int_{0}^{1}{\left[u \tan^{-1}(u) \right]du}}$

3. We will apply the method of integration by parts

(a) Assigning first and second functions:

   ♦ Let first function be: $\small{f(u)=\tan^{-1}u}$

   ♦ Let second function be: $\small{g(u)=u}$

(b) Finding A:

$\small{A~=~\int{\left[g(u) \right]dx}~=~\frac{u^2}{2}}$

(c) $\small{\big[f(u) \left(A \right) \big]~=~\big[ \tan^{-1}u\big] \,\big[\frac{u^2}{2} \big]~=~\big[\frac{u^2\,\tan^{-1}u}{2} \big]}$

• This is the first term.

(d) $\small{f'(u)~=~\frac{1}{1 + u^2}}$

(e) $\small{\int{\big[f'(u)\,\left(A \right)  \big]du}~=~\int{\big[\frac{1}{1 + u^2}\,\left(\frac{u^2}{2} \right)\big]du}}$

$\small{~=~\int{\big[\frac{u^2}{2(1 + u^2)}\big]du}~=~\frac{1}{2}\int{\big[\frac{u^2}{1 + u^2}\big]du}}$

$\small{~=~\frac{1}{2}\int{\big[\frac{u^2 + 1 - 1}{1 + u^2}\big]du}~=~\frac{1}{2}\int{\big[1~-~\frac{1}{1 + u^2}\big]du}}$

$\small{~=~\frac{1}{2}\int{\big[1\big]du}~-~\frac{1}{2}\int{\big[\frac{1}{1 + u^2}\big]du}}$

$\small{~=~\frac{u}{2}~-~\frac{\tan^{-1}u}{2}}$

• This is the second term.

(f) So we get:

$\small{\int_{0}^{\frac{1}{2}}{\left[u \tan^{-1}(u) \right]du}~=~\text{First term - Second term}}$

$\small{~=~\big[\frac{u^2\,\tan^{-1}u}{2}\big]~-~\big[\frac{u}{2}~-~\frac{\tan^{-1}u}{2} \big]}$

$\small{~=~\frac{u^2\,\tan^{-1}u}{2}~-~\frac{u}{2}~+~\frac{\tan^{-1}u}{2}}$

$\small{~=~\frac{(u^2 + 1)(\tan^{-1}u)~-~u}{2}}$

4. So from (2), we get:

$\small{I= 2 \int_{0}^{1}{\left[u \tan^{-1}(u) \right]du}~=~2 \big[\frac{(u^2 + 1)(\tan^{-1}u)~-~u}{2} \big]_0^1}$

$\small{~=~ \big[(u^2 + 1)(\tan^{-1}u)~-~u \big]_0^1}$

$\small{~=~ \big[(1^2 + 1)(\tan^{-1}1)~-~1 \big]~-~\big[(0 + 1)(\tan^{-1}0)~-~0 \big]}$

$\small{~=~ \big[(2)(\pi/4)~-~1 \big]~-~\big[(1)(0)~-~0 \big]}$

$\small{~=~ \frac{\pi}{2}~-~1}$

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The link below gives a few more miscellaneous examples:

Miscellaneous Exercise

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In the next chapter, we will see applications of integrals.

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23.30 - Miscellaneous Examples (2) on Integrals

In the previous section, we saw some miscellaneous examples on integrals. In this section, we will see a few more miscellaneous examples.

Solved Example 23.133
Integrate $\small{\frac{1}{\sqrt{x+a}~+~\sqrt{x+b}}}$    
Solution:
1. First we will rearrange the given expression:
$\small{\frac{1}{\sqrt{x+a}~+~\sqrt{x+b}}
~=~\frac{1}{\sqrt{x+a}~+~\sqrt{x+b}}\left(\frac{\sqrt{x+a}~-~\sqrt{x+b}}{\sqrt{x+a}~-~\sqrt{x+b}} \right)}$

$\small{~=~\frac{\sqrt{x+a}~-~\sqrt{x+b}}{(x+a)~-~(x+b)}~=~\frac{\sqrt{x+a}~-~\sqrt{x+b}}{a-b}}$

2. So we want: $\small{I = \int{\left[\frac{\sqrt{x+a}~-~\sqrt{x+b}}{a-b}\right]dx}}$

$\small{~=~\frac{1}{a-b}\int{\left[\sqrt{x+a}~-~\sqrt{x+b}\right]dx}}$

$\small{~=~\frac{1}{a-b}\int{\left[\sqrt{x+a}\right]dx}~-~\frac{1}{a-b}\int{\left[\sqrt{x+b}\right]dx}}$

3. This integration gives:

$\small{\frac{1}{a-b}\left[\frac{(x+a)^{3/2}}{3/2} \right]~-~\frac{1}{a-b}\left[\frac{(x+b)^{3/2}}{3/2} \right]}$

$\small{~=~\frac{2}{3(a-b)}\left[(x+a)^{3/2}~-~(x+b)^{3/2} \right]}$

Solved Example 23.134
Integrate $\small{\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}}$
Solution:
1. First we will rearrange the given expression:

Let $\small{\sqrt x = \cos(2 \theta)}$. Then we get:

$\small{\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}
~=~\sqrt{\frac{1-\cos(2 \theta)}{1+\cos(2 \theta)}}~=~\sqrt{\tan^2(\theta)}~=~\tan \theta}$

Also, since $\small{\sqrt x = \cos(2 \theta)}$, we can write:

$\small{\frac{1}{2 \sqrt x} \frac{dx}{d\theta}~=~-2\sin (2\theta)}$

$\small{\Rightarrow dx~=~-4 \sqrt x\, \sin(2 \theta) d \theta}$

$\small{\Rightarrow dx~=~-4 \cos(2\theta)\, \sin(2 \theta) d \theta}$

2. So we want: $\small{I = \int{\left[\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\right]dx}= \int{\left[\tan \theta\right]dx}}$

$\small{~=~\int{\left[\tan \theta\right][-4 \cos(2\theta)\, \sin(2 \theta) d \theta]}}$

$\small{~=~(-4)\int{\left[\tan \theta \,\cos(2\theta)\, \sin(2 \theta)\right]d \theta}}$

$\small{~=~(-4)\int{\left[\frac{\sin \theta}{\cos \theta} \,\cos(2\theta)\, 2 \sin \theta \cos \theta\right]d \theta}}$

$\small{~=~(-4)\int{\left[2 \sin^2 \theta \,\cos(2\theta) \right]d \theta}}$

$\small{~=~(-4)\int{\left[[1- \cos(2\theta)] \,\cos(2\theta) \right]d \theta}}$

$\small{~=~(-4)\int{\left[\cos(2\theta)- \cos^2(2\theta) \right]d \theta}}$

$\small{~=~(-4)\int{\left[\cos(2\theta)- \left(\frac{1+\cos(4 \theta)}{2} \right) \right]d \theta}}$

$\small{~=~(-4)\int{\left[\cos(2\theta) \right]d \theta}~-~(-4)\int{\left[\frac{1}{2}  \right]d \theta}~-~(-4)\int{\left[\frac{\cos(4 \theta)}{2}  \right]d \theta}}$

$\small{~=~(-4)\int{\left[\cos(2\theta) \right]d \theta}~+~(2)\int{\left[1  \right]d \theta}~+~(2)\int{\left[\cos(4 \theta) \right]d \theta}}$

3. This integration gives:

$\small{I~=~(-4)\left[\frac{\sin(2\theta)}{2} \right]~+~(2)\left[\theta  \right]~+~(2)\left[\frac{\sin(4\theta)}{4} \right]}$

$\small{~=~(-2)\left[\sin(2\theta) \right]~+~2 \theta~+~\frac{\sin(4\theta)}{2}}$

$\small{~=~(-2)\left[\sin(2\theta) \right]~+~2 \theta~+~\sin(2 \theta) \cos (2 \theta)}$

4. since $\small{\sqrt x = \cos(2 \theta)}$, we can write:

$\small{2 \theta~=~\cos^{-1}\left(\sqrt x \right)}$

$\small{\Rightarrow \theta~=~\frac{\cos^{-1}\left(\sqrt x \right)}{2}}$

$\small{\Rightarrow \sin (2\theta)~=~\sqrt {1 - \cos^2(2\theta)}~=~\sqrt{1-x}}$

5. Substituting the above results in (3), we get:

$\small{I~=~(-2)\left[\sin(2\theta) \right]~+~2 \theta~+~\sin(2 \theta) \cos (2 \theta)}$

$\small{~=~(-2)\left[\sqrt{1-x}\right]~+~\cos^{-1}\left(\sqrt x \right)~+~\left[\sqrt{1-x}\, \sqrt{x} \right]}$

$\small{~=~(-2)\left[\sqrt{1-x}\right]~+~\cos^{-1}\left(\sqrt x \right)~+~\left[\sqrt{x-x^2} \right]~+~\rm{C}}$

Solved Example 23.135
Integrate $\small{\frac{\sin^{-1} \sqrt{x} - \cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x}},~x \in [0,1]}$
Solution:
1. First we will rearrange the given expression:

Let $\small{\sin^{-1} \sqrt x = \theta}$. Then we get:

$\small{\sin \theta~=~\sqrt x}$

$\small{\Rightarrow \cos\left(\frac{\pi}{2} - \theta\right)~=~\sqrt x}$

$\small{\Rightarrow \cos^{-1} \sqrt x ~=~\frac{\pi}{2} - \theta}$

• Also, since $\small{\sin \theta~=~\sqrt x}$, we can write:

$\small{\cos \theta \frac{d \theta}{dx}~=~\frac{1}{2 \sqrt x}}$

$\small{\Rightarrow dx~=~2 \sqrt x \cos \theta \,d\theta}$

$\small{\Rightarrow dx~=~2 \sin\theta \cos \theta \,d\theta}$

$\small{\Rightarrow dx~=~\sin(2\theta) \,d\theta}$

2. So we want: $\small{I = \int{\left[\frac{\sin^{-1} \sqrt{x} ~-~ \cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} ~+~ \cos^{-1} \sqrt{x}}\right]dx}}$

$\small{~=~\int{\left[\frac{\theta~-~\left(\frac{\pi}{2} - \theta\right)}{\theta~+~\left(\frac{\pi}{2} - \theta\right)} \right][\sin(2\theta) \,d\theta]}}$

$\small{~=~\int{\left[\frac{2\theta~-~\frac{\pi}{2}}{\frac{\pi}{2}} \right][\sin(2\theta) \,d\theta]}}$

$\small{~=~\int{\left[\frac{4 \theta}{\pi} ~-~1\right][\sin(2\theta) \,d\theta]}}$

$\small{~=~\int{\left[\frac{4 \theta}{\pi}\right][\sin(2\theta) \,d\theta]}~-~\int{\left[1\right][\sin(2\theta) \,d\theta]}}$

$\small{~=~\int{\left[\frac{4 \theta}{\pi} \sin(2\theta)\right]d\theta}~-~\int{\left[\sin(2\theta)\right]d\theta}}$

$\small{~=~\frac{4}{\pi}\int{\left[\theta\, \sin(2\theta)\right]d\theta}~-~\int{\left[\sin(2\theta)\right]d\theta}}$

$\small{~=~\left(\frac{4}{\pi} \right)I_1~-~I_2}$

3. Next we will calculate I₁:

(a) Assigning first and second functions:

   ♦ Let first function be: $\small{f(\theta)=\theta}$

   ♦ Let second function be: $\small{g(\theta)=\sin(2\theta)}$

(b) Finding A:

$\small{A~=~\int{\left[g(\theta) \right]dx}~=~\int{\left[\sin(2\theta)\right]d\theta}~=~\frac{- \cos (2 \theta)}{2}}$

(c) $\small{\big[f(\theta) \left(A \right) \big]~=~\big[\frac{-\theta \cos (2 \theta)}{2} \big]}$

• This is the first term.

(d) $\small{f'(\theta)~=~1}$

(e) $\small{\int{\big[f'(\theta)\,\left(A \right)  \big]d\theta}~=~\int{\big[1\,\left(\frac{- \cos (2 \theta)}{2} \right)\big]d\theta}}$

$\small{~=~\frac{-1}{2}\int{\big[\cos(2\theta)\big]d\theta}~=~\frac{-\sin(2\theta)}{4}}$

• This is the second term.

(f) So we get:

$\small{I_1~=~\text{First term - Second term}}$

$\small{~=~\big[\frac{-\theta \cos (2 \theta)}{2}\big]~-~\big[\frac{-\sin(2\theta)}{4}\big]~=~\frac{\sin(2\theta)~-~2 \theta \cos(2 \theta)}{4}}$

4. Next we calculate I₂:

$\small{\int{\left[\sin(2\theta)\right]d\theta}~=~\frac{-\cos(2\theta)}{2}}$

5. So from (2), we get: $\small{I = \left(\frac{4}{\pi} \right)I_1 - I_2}$

$\small{~=~\left(\frac{4}{\pi} \right)\frac{\sin(2\theta)~-~2 \theta \cos(2 \theta)}{4}~-~\frac{-\cos(2\theta)}{2}}$

$\small{~=~\frac{\sin(2\theta)~-~2 \theta \cos(2 \theta)}{\pi}~+~\frac{\cos(2\theta)}{2}}$

6. Now we can substitute for $\small{\theta}$.

• Since $\small{\sin \theta~=~\sqrt x}$, we can write:

$\small{\cos \theta ~=~\sqrt{1-x}}$

$\small{\Rightarrow \sin(2 \theta) ~=~2 \sin \theta\,\cos \theta~=~2\sqrt{x(1-x)}~=~2\sqrt{(x-x^2)}}$

$\small{\Rightarrow \cos(2 \theta) ~=~\cos^2\theta~-~\sin^2 \theta~=~1-2x}$

7. Based on the result in (5), we get:

$\small{I~=~\frac{\sin(2\theta)~-~2 \theta \cos(2 \theta)}{\pi}~+~\frac{\cos(2\theta)}{2}}$

$\small{~=~\frac{2\sqrt{(x-x^2)}~-~\left[2 \sin^{-1}\sqrt{x} \right] (1-2x)}{\pi}~+~\frac{1-2x}{2}}$

$\small{~=~\frac{2\sqrt{(x-x^2)}~+~\left[2 \sin^{-1}\sqrt{x} \right] (2x-1)}{\pi}~+~\frac{1-2x}{2}}$

$\small{~=~\frac{2\sqrt{(x-x^2)}~+~\left[2 \sin^{-1}\sqrt{x} \right] (2x-1)}{\pi}~+~\frac{1}{2}~-~x~+~\rm{C_1}}$

$\small{~=~\frac{2\sqrt{(x-x^2)}~+~\left[2 \sin^{-1}\sqrt{x} \right] (2x-1)}{\pi}~-~x~+~\rm{C}}$

Solved Example 23.136
Integrate $\small{\frac{\sin x}{\sin(x-a)}}$
Solution:
1. First we will rearrange the given expression:

Let $\small{u = x-a}$. Then we get:

$\small{x~=~u+a}$

Also, $\small{\frac{du}{dx}~=~1 \Rightarrow du = dx}$

2. So we want: $\small{I = \int{\left[\frac{\sin x}{\sin(x-a)}\right]dx} = \int{\left[\frac{\sin (u+a)}{\sin(u)}\right]du}}$

$\small{= \int{\left[\frac{\sin u \cos a~+~\cos u \sin a}{\sin u}\right]du}= \int{\left[\cos a~+~\cot u \sin a\right]du}}$

$\small{= \int{\left[\cos a\right]du}~+~\int{\left[\cot u \sin a\right]du}}$

$\small{= \cos a \int{\left[1\right]du}~+~\sin a \int{\left[\cot u\right]du}}$

$\small{= I_1~+~I_2}$

3. Next we calculate I₁. We get:

$\small{= \cos a \int{\left[1\right]du}~=~\cos a \,(u)}$

4. Next we calculate I₂:

$\small{\sin a \int{\left[\cot u\right]du}~=~ (\sin a)\log \left|\sin u \right|}$

5. So we can write:

$\small{I=I_1 + I_2~=~\cos a \,(u)~+~(\sin a)\log \left|\sin u \right|}$

• Substituting for u, we get:

$\small{I~=~\cos a \,(x-a)~+~(\sin a)\log \left|\sin (x-a) \right|~+~\rm{C_1}}$

$\small{~=~x \cos a ~-~ a \cos a~+~(\sin a)\log \left|\sin (x-a) \right|~+~\rm{C_1}}$

$\small{~=~x \cos a~+~(\sin a)\log \left|\sin (x-a) \right|~+~\rm{C}}$

Solved Example 23.137
Integrate $\small{\frac{\sin^8 x~-~\cos^8 x}{1~-~2 \sin^2 x \cos^2 x}}$
Solution:
1. First we will rearrange the given expression:

$\small{\frac{(\sin^4 x)^2~-~(\cos^4 x)^2}{1~-~2 \sin^2 x \cos^2 x}~=~\frac{\left[(\sin^4 x)~+~(\cos^4 x)\right] \left[(\sin^4 x)~-~(\cos^4 x)\right]}{1~-~2 \sin^2 x \cos^2 x}}$

$\small{~=~\frac{\left[(\sin^2 x)^2~+~(\cos^2 x)^2 ~+~2 \sin^2 x \cos^2 x  ~-~2 \sin^2 x \cos^2 x\right] \left[(\sin^4 x)~-~(\cos^4 x)\right]}{1~-~2 \sin^2 x \cos^2 x}}$

$\small{~=~\frac{\left[[(\sin^2 x)~+~(\cos^2 x)]^2  ~-~2 \sin^2 x \cos^2 x\right] \left[(\sin^4 x)~-~(\cos^4 x)\right]}{1~-~2 \sin^2 x \cos^2 x}}$

$\small{~=~\frac{\left[[1]^2  ~-~2 \sin^2 x \cos^2 x\right] \left[(\sin^4 x)~-~(\cos^4 x)\right]}{1~-~2 \sin^2 x \cos^2 x}}$

$\small{~=~\left[(\sin^4 x)~-~(\cos^4 x)\right]}$

$\small{~=~\left[(\sin^2 x)~+~(\cos^2 x)\right]~\left[(\sin^2 x)~-~(\cos^2 x)\right]}$

$\small{~=~\left[1\right]~\left[(\sin^2 x)~-~(\cos^2 x)\right]}$

$\small{~=~\left[-1\right]~\left[(\cos^2 x)~-~(\sin^2 x)\right]}$

$\small{~=~- \cos(2x)}$

2. So we get: $\small{I = \int{\left[-\cos(2x)\right]dx} = \frac{-\sin(2x)}{2}~+~\rm{C}}$

Solved Example 23.138
Integrate $\small{\frac{1}{\cos (x+a)\, \cos(x+b)}}$
Solution:
1. First we will rearrange the given expression:

$\small{\frac{1}{\cos (x+a)\, \cos(x+b)}~=~\frac{\sin(a-b)}{\sin(a-b)}\times\frac{1}{\cos (x+a)\, \cos(x+b)}}$

$\small{~=~\frac{1}{\sin(a-b)}\times\frac{\sin(a-b)}{\cos (x+a)\, \cos(x+b)}~=~\frac{1}{\sin(a-b)}\times\frac{\sin(a-b-x+x)}{\cos (x+a)\, \cos(x+b)}}$

$\small{~=~\frac{1}{\sin(a-b)}\times\frac{\sin\left[ (x+a)-(x+b)\right]}{\cos (x+a)\, \cos(x+b)}}$

$\small{~=~\frac{1}{\sin(a-b)}\times\frac{\sin(x+a) \cos(x+b)~-~\cos(x+a) \sin(x+b)}{\cos (x+a)\, \cos(x+b)}}$

$\small{~=~\frac{1}{\sin(a-b)}\times\left[\frac{\sin(x+a) \cos(x+b)}{\cos (x+a)\, \cos(x+b)}~-~\frac{\cos(x+a) \sin(x+b)}{\cos (x+a)\, \cos(x+b)} \right]}$

$\small{~=~\frac{1}{\sin(a-b)}\times\left[\frac{\sin(x+a)}{\cos (x+a)}~-~\frac{ \sin(x+b)}{\cos(x+b)} \right]}$

$\small{~=~\frac{1}{\sin(a-b)}\times\left[\tan(x+a)~-~\tan(x+b) \right]}$

2. So we get:

$\small{I = \int{\left[\frac{1}{\cos (x+a)\, \cos(x+b)}\right]dx} = \int{\left[\frac{1}{\sin(a-b)}\times\left[\tan(x+a)~-~\tan(x+b) \right]\right]dx}}$

$\small{= \frac{1}{\sin(a-b)} \int{\left[\tan(x+a)~-~\tan(x+b) \right]dx}}$

$\small{= \frac{1}{\sin(a-b)} \left[\log \left|\sec(x+a) \right|~-~\log \left|\sec(x+b) \right| \right]~+~\rm{C}}$

$\small{= \frac{1}{\sin(a-b)} \left[\log \left| \frac{\sec(x+a)}{\sec(x+b)}  \right| \right]~+~\rm{C}}$

$\small{= \frac{1}{\sin(a-b)} \left[\log \left| \frac{\cos(x+b)}{\cos(x+a)}  \right| \right]~+~\rm{C}}$

Solved Example 23.139
Integrate $\small{\frac{1}{\sqrt{\sin^3 x \sin(x+\alpha)}}}$
Solution:
1. First we will rearrange the portion inside square root symbol:

$\small{\sin^3 x \sin(x+\alpha)}$

$\small{~=~\sin^3 x \left[\sin x \cos \alpha~+~\cos x \sin\alpha \right]}$

$\small{~=~\sin^4 x \cos \alpha~+~\sin^3 x\cos x \sin\alpha }$

$\small{~=~\sin^4 x \cos \alpha~+~\sin^3 x\cos x \sin\alpha \frac{\sin x}{\sin x}}$

$\small{~=~\sin^4 x \cos \alpha~+~\sin^4 x\cot x \sin\alpha}$

$\small{~=~\sin^4 x \left[\cos \alpha~+~\cot x \sin\alpha \right]}$

2. So we get:

$\small{I = \int{\left[\frac{1}{\sqrt{\sin^3 x \sin(x+\alpha)}}\right]dx} =\int{\left[\frac{1}{\sqrt{\sin^4 x \left[\cos \alpha~+~\cot x \sin\alpha \right]}}\right]dx}}$

$\small{=\int{\left[\frac{1}{\sin^2 x \sqrt{\cos \alpha~+~\cot x \sin\alpha }}\right]dx}=\int{\left[\frac{\csc^2 x}{\sqrt{\cos \alpha~+~\cot x \sin\alpha }}\right]dx}}$

3. Let us put $\small{u = \cos \alpha~+~\cot x\, \sin\alpha}$

• Then $\small{\frac{du}{dx}~=~0 + \sin\alpha(-\csc^2 x)~=~-\sin \alpha\,\csc^2 x}$

$\small{\Rightarrow -\sin \alpha\,\csc^2 x\,dx~=~du}$

4. So we want:

$\small{I=\int{\left[\frac{\csc^2 x}{\sqrt{\cos \alpha~+~\cot x \sin\alpha }}\right]dx}=\int{\left[\frac{(-\sin \alpha)\csc^2 x}{(-\sin \alpha)\sqrt{\cos \alpha~+~\cot x \sin\alpha }}\right]dx}}$

$\small{=\int{\left[\frac{1}{(-\sin \alpha)\sqrt{u}}\right]du}=\frac{1}{(-\sin \alpha)}\int{\left[\frac{1}{\sqrt{u}}\right]du}}$

5. This integration gives:

$\small{\frac{1}{(-\sin \alpha)}\left[\frac{u^{1/2}}{1/2}~+~\rm{C_1}\right]~=~\frac{-2\,u^{1/2}}{\sin\alpha}~+~\rm{C}}$

6. Substituting for u, we get:

$\small{I~=~\frac{-2\,(\cos \alpha~+~\cot x\, \sin\alpha)^{1/2}}{\sin\alpha}~+~\rm{C}}$

7. From step (1), we have:

$\small{\sin^3 x \sin(x+\alpha)~=~\sin^4 x \left[\cos \alpha~+~\cot x \sin\alpha \right]}$

$\small{\Rightarrow \sin(x+\alpha)~=~\sin x \left[\cos \alpha~+~\cot x \sin\alpha \right]}$

$\small{\Rightarrow \cos \alpha~+~\cot x \sin\alpha~=~\frac{\sin(x+\alpha)}{\sin x}}$

• So from (6), we get:

$\small{I~=~\frac{-2}{\sin\alpha} \sqrt{\frac{\sin(x+\alpha)}{\sin x}}~+~\rm{C}}$

Solved Example 23.140
Integrate $\small{\tan^{-1}\sqrt{\frac{1-x}{1+x}}}$
Solution:
1. First we will rearrange the given expression:

Let $\small{x = \cos(2 \theta)}$. Then we get:

$\small{\sqrt{\frac{1-x}{1+ x}}
~=~\sqrt{\frac{1-\cos(2 \theta)}{1+\cos(2 \theta)}}~=~\sqrt{\tan^2(\theta)}~=~\tan \theta}$

• Also, since $\small{x = \cos(2 \theta)}$, we can write:

$\small{\frac{dx}{d\theta}~=~-2\sin (2\theta)}$

$\small{\Rightarrow dx~=~-2 \sin(2 \theta) d \theta}$

• Also, since $\small{x = \cos(2 \theta)}$, we can write:

$\small{2\theta~=~\cos^{-1}x ~~\text{and}~~ \sin(2\theta)~=~\sqrt{1 - x^2}}$

2. So we want: $\small{I = \int{\left[\tan^{-1}\sqrt{\frac{1-x}{1+x}}\right]dx}= \int{\left[\tan^{-1}(\tan \theta)\right]dx}}$

$\small{~=~ \int{\left[\theta\right]dx}~=~\int{\left[\theta\right]}(-2 \sin(2 \theta) d \theta)~=~-2\int{\left[\theta \sin(2 \theta)\right]}d \theta}$

$\small{\int{\left[\theta \sin(2 \theta)\right]}d \theta}$ is already done in solved example 23.135 above.

So we get:

$\small{I~=~-2\left[\frac{\sin(2\theta)~-~2 \theta \cos(2 \theta)}{4}~+~\rm{C_1}\right]~=~(-1)\left[\frac{\sin(2\theta)~-~2 \theta \cos(2 \theta)}{2}~+~\rm{C_2}\right]}$

$\small{~=~\frac{1}{2}\left[2 \theta \cos(2 \theta)~-~\sin(2\theta)~+~\rm{C_3}\right]}$

3. Substituting for $\small{2\theta,~ \cos(2 \theta)~\rm{and}~\sin(2 \theta),}$ we get:

$\small{I~=~\frac{1}{2}\left[\cos^{-1}x\,(x)~-~\sqrt{1 - x^2}~+~\rm{C_3}\right]}$

$\small{\Rightarrow I~=~\frac{1}{2}\left[(x) \cos^{-1}x\,~-~\sqrt{1 - x^2}\right]~+~\rm{C}}$

Solved Example 23.141
Integrate $\small{\frac{\cos(2x)}{(\sin x + \cos x)^2}}$
Solution:
1. First we will rearrange the given expression:

$\small{\frac{\cos(2x)}{(\sin x + \cos x)^2}
~=~\frac{\cos(2x)}{\sin^2 x +2 \sin x \cos x + \cos^2 x}~=~\frac{\cos(2x)}{1 +2 \sin x \cos x}~=~\frac{\cos(2x)}{1 + \sin (2x)}}$

• The derivative of [1+sin(2x)] is cos(2x).

• So we put $\small{u = 1+\sin(2x)\Rightarrow \frac{du}{dx}~=~2 \cos (2x)}$

$\small{\Rightarrow \frac{du}{dx}~=~2 \cos (2x) \Rightarrow du~=~2 \cos(2x) dx}$

2. So we want: $\small{I = \int{\left[\frac{\cos(2x)}{(\sin x + \cos x)^2}\right]dx}= \int{\left[\frac{2\cos(2x)}{2(1 + \sin (2x))}\right]dx}}$

$\small{~=~ \int{\left[\frac{1}{2(u)} \right]du}~=~\frac{1}{2} \int{\left[\frac{1}{u} \right]du}}$

3. This integration gives:

$\small{I~=~\frac{1}{2} \left[\log \left|u \right|~+~\rm{C_1}\right]~=~\frac{1}{2} \log \left|u \right|~+~\rm{C}}$

4. Substituting for u, we get:

$\small{I~=~\frac{1}{2} \log \left|1 + \sin (2x) \right|~+~\rm{C}}$

$\small{~=~\frac{1}{2} \log \left|(\sin x + \cos x)^2 \right|~+~\rm{C}}$

$\small{~=~\frac{2}{2} \log \left|(\sin x + \cos x) \right|~+~\rm{C}}$

$\small{~=~ \log \left|(\sin x + \cos x) \right|~+~\rm{C}}$

Solved Example 23.142
Integrate $\small{\frac{5x}{(x+1)(x^2 + 9)}}$
Solution:
1. The numerator is a polynomial of degree 1. The denominator is a polynomial of degree 3.

2. So it is a proper rational function. We can straight away start partial fraction decomposition

3. The denominator is already factorized:
$\small{(x+1)(x^2 + 9)}$

   ♦ One factor is quadratic.
   
   ♦ That quadratic factor cannot be further factorized.
   
   ♦ All other factors are linear.

4. The quadratic factor $\small{(x^2 + 9)}$ cannot be further factorized. Then we are able to write:

$\small{\frac{5x}{(x+1)(x^2 + 9)}~=~\left[\frac{Ax + B}{x^2 + 9}\right]~+~\frac{A_1}{x+1}}$
Where A, B and A₁ are real numbers.

5. To find A, B and A₁, we make denominators same on both sides:

$\small{\frac{5x}{(x+1)(x^2 + 9)}~=~\frac{(Ax+B) (x+1)~+~A_1 (x^2+9)}{(x+1)(x^2 + 9)}}$

6. Since denominators are same on both sides, we can equate the numerators. We get:

$\small{5x~=~(Ax+B) (x+1)~+~A_1 (x^2+9)}$

7. After equating the numerators, we can use suitable substitution.

   ♦ Put x = −1. We get: $\small{-5~=~10A_1}$. So $\small{A_1 = \frac{-1}{2}}$  
    
   ♦ Put x = 0. We get: $\small{0~=~B~-~(9/2)}$. So $\small{B = \frac{9}{2}}$  
     
   ♦ Put x = 1. We get: $\small{5~=~(A+9/2) (2)~-~5}$. So $\small{A = \frac{1}{2}}$
   
8. Now the result in (4) becomes:

$\small{\frac{5x}{(x+1)(x^2 + 9)}~=~\left[\frac{(1/2)x + 9/2}{x^2 + 9}\right]~-~\frac{1/2}{x+1}}$

$\small{~=~\left[\frac{x ~+~ 9}{2(x^2 + 9)}\right]~-~\frac{1}{2(x + 1)}}$

9. So the integration becomes easy. We get:

$\small{\frac{1}{4} \log (x^2+9)~+~\frac{3}{2} \tan^{-1}\left(\frac{x}{3} \right)~-~\frac{1}{2} \log \left|x+1 \right|~+~\rm{C}}$

• The reader may write all the steps involved in the integration process.

Solved Example 23.143
Integrate $\small{\frac{1}{(x^2 + 1)(x^2 + 4)}}$
Solution:
1. The numerator is a polynomial of degree 0. The denominator is a polynomial of degree 4.

2. So it is a proper rational function. We can straight away start partial fraction decomposition

3. The denominator is already factorized:
$\small{(x^2 + 1)(x^2 + 4)}$

   ♦ Both factors are quadratic.
   
   ♦ Both quadratic factors cannot be further factorized.
   
4. Then we are able to write:

$\small{\frac{1}{(x^2 + 1)(x^2 + 4)}~=~\left[\frac{A_1 x + B_1}{x^2 + 1}\right]~+~\left[\frac{A_2 x + B_2}{x^2 + 4}\right]}$
Where A₁, B₁, A₂ and B₂ are real numbers.

5. To find those real numbers, we make denominators same on both sides:

$\small{\frac{1}{(x^2 + 1)(x^2 + 4)}~=~\frac{(A_1 x + B_1) (x^2 + 4)~+~(A_2 x + B_2) (x^2 + 1)}{(x+1)(x^2 + 4)}}$

6. Since denominators are same on both sides, we can equate the numerators. We get:

$\small{1~=~(A_1 x + B_1) (x^2 + 4)~+~(A_2 x + B_2) (x^2 + 1)}$

7. After equating the numerators, we can use suitable substitution.

   ♦ Put x = 0. We get: $\small{1~=~4B_1~+~B_2}$
    
   ♦ Put x = −1. We get: $\small{1~=~-5A_1~+~5B_1~-~2A_2~+~2B_2}$
     
   ♦ Put x = −2. We get: $\small{1~=~-16A_1~+~8B_1~-~10A_2~+~5B_2}$
   
   ♦ Put x = 1. We get: $\small{1~=~5A_1~+~5B_1~+~2A_2~+~2B_2}$
   
• Solving the four equations, we get:

$\small{A_1 = 0,~A_2=0,~B_1=\frac{1}{3},~\rm{and}~B_2=\frac{-1}{3}}$
   
8. Now the result in (4) becomes:

$\small{\frac{1}{(x^2 + 1)(x^2 + 4)}~=~\left[\frac{1}{3(x^2 + 1)}\right]~-~\left[\frac{1}{3(x^2 + 4)}\right]}$

9. So the integration becomes easy. We get:

$\small{\frac{1}{3} \tan^{-1}\left(x \right)~-~\frac{1}{6} \tan^{-1}\left(\frac{x}{2} \right)~+~\rm{C}}$

• The reader may write all the steps involved in the integration process.

Solved Example 23.144
Integrate $\small{\frac{x^2 + x + 1}{(x + 1)^2 (x + 2)}}$
Solution:
1. The numerator is a polynomial of degree 2. The denominator is a polynomial of degree 3.

2. So it is a proper rational function. We can straight away start partial fraction decomposition

3. The denominator is already factorized:
$\small{(x + 1)^2 \,(x + 2)}$

   ♦ All factors are linear.
   
   ♦ The factor (x+1) appears twice.
   
4. Then we are able to write:

$\small{\frac{x^2 + x + 1}{(x + 1)^2 \, (x + 2)}~=~\left[\frac{A_1}{x + 1}\right]~+~\left[\frac{A_2}{(x + 1)^2}\right]~+~\left[\frac{A_3}{x + 2}\right]}$
Where A₁, A₂ and A₃ are real numbers.

5. To find those real numbers, we make denominators same on both sides:

$\small{\frac{x^2 + x + 1}{(x + 1)^2 \, (x + 2)}~=~\frac{A_1 (x + 1)(x+2)~+~A_2 (x + 2)~+~A_3 (x+1)^2}{(x + 1)^2 \, (x + 2)}}$

6. Since denominators are same on both sides, we can equate the numerators. We get:

$\small{x^2 + x + 1~=~A_1 (x + 1)(x+2)~+~A_2 (x + 2)~+~A_3 (x+1)^2}$

7. After equating the numerators, we can use suitable substitution.

   ♦ Put x = 0. We get: $\small{1~=~2 A_1~+~2 A_2~+~A_3}$
    
   ♦ Put x = −1. We get: $\small{1~=~A_2}$
     
   ♦ Put x = −2. We get: $\small{3~=~A_3}$
   
• Solving the three equations, we get:

$\small{A_1 = -2,~A_2=1,~\rm{and}~A_3=3}$
   
8. Now the result in (4) becomes:

$\small{\frac{x^2 + x + 1}{(x + 1)^2 \, (x + 2)}~=~\left[\frac{-2}{x + 1}\right]~+~\left[\frac{1}{(x + 1)^2}\right]~+~\left[\frac{3}{x + 2}\right]}$

9. So the integration becomes easy. We get:

$\small{-2 \log \left|x+1 \right|~-~\frac{1}{x+1}~+~3 \log \left|x+2 \right|~+~\rm{C}}$

• The reader may write all the steps involved in the integration process.

Solved Example 23.145
Integrate $\small{\frac{e^{5 \log x}~-~e^{4 \log x}}{e^{3 \log x}~-~e^{2 \log x}}}$
Solution:
1. Consider the first term of the numerator. Let us write:

$\small{u~=~e^{5 \log x}}$

• Taking log on both sides, we get: $\small{\log u~=~\log\left(e^{5 \log x} \right)}$

$\small{\Rightarrow \log u~=~5 \log x\left[\log\left(e \right) \right]~=~5 \log(x)~=~\log\left(x^5 \right)}$

$\small{\Rightarrow u~=~x^5}$

2. In this way, all terms can be rearranged. The given expression becomes:

$\small{\frac{x^5~-~x^4}{x^3~-~x^2}}$. This can be rearranged as: $\small{\frac{x^4 \left(x~-~1 \right)}{x^2 \left(x~-~1 \right)}~=~x^2}$

3. So we want: $\small{I = \int{\left[\frac{e^{5 \log x}~-~e^{4 \log x}}{e^{3 \log x}~-~e^{2 \log x}}\right]dx}= \int{\left[x^2 \right]dx}}$

4. This integration gives: $\small{I~=~\frac{x^3}{3}~+~\rm{C}}$

Solved Example 23.146
Integrate $\small{\cos^3 x\,e^{\log(\sin x)}}$
Solution:
1. First we will rearrange the second portion:

• Let us write:

$\small{u~=~e^{\log(\sin x)}}$

• Taking log on both sides, we get: $\small{\log u~=~\log\left(e^{\log(\sin x)} \right)}$

$\small{\Rightarrow \log u~=~\log(\sin x) \left[\log\left(e \right) \right]~=~\log(\sin x)}$

$\small{\Rightarrow u~=~\sin x}$

2. So the given expression becomes:

$\small{\cos^3 x\,\sin x}$.

3. So we want: $\small{I = \int{\left[\cos^3 x\,\sin x\right]dx}}$

4. Put $\small{t~=~\cos x}$

• Then we get:$\small{\frac{dt}{dx}~=~-\sin x \Rightarrow -\sin x \, dx~=~dt}$

5. So we want:

$\small{I = \int{\left[\cos^3 x\,\sin x\right]dx} = \int{\left[(-1)(-1)\cos^3 x\,\sin x\right]dx} = (-1)\int{\left[t^3 \right]dt}}$

6. This integration gives: $\small{I~=~(-1)\left[\frac{t^4}{4} ~+~\rm{C_1} \right]~=~(-1)\frac{t^4}{4} ~+~\rm{C}}$

7. Substituting for t, we get:

$\small{I~=~(-1)\frac{\cos^4 x}{4} ~+~\rm{C}}$

Solved Example 23.147
Integrate $\small{e^{3 \log(x)}\,(x^4 + 1)^{-1}}$
Solution:
1. First we will rearrange $\small{\left[e^{3 \log(x)} \right]}$:

• Let us write:

$\small{u~=~e^{3 \log(x)}}$

• Taking log on both sides, we get: $\small{\log u~=~\log\left(e^{3 \log(x)} \right)}$

$\small{\Rightarrow \log u~=~3 \log(x) \left[\log\left(e \right) \right]~=~3 \log(x)~=~\log\left(x^3 \right)}$

$\small{\Rightarrow u~=~x^3}$

2. So the given expression becomes:

$\small{x^3\,(x^4 + 1)^{-1}~=~\frac{x^3}{x^4 + 1}}$

3. So we want: $\small{I = \int{\left[e^{3 \log(x)}\,(x^4 + 1)^{-1}\right]dx}= \int{\left[\frac{x^3}{x^4 + 1}\right]dx}}$

4. Put $\small{t~=~x^4 + 1}$

• Then we get:$\small{\frac{dt}{dx}~=~4x^3 \Rightarrow  4x^3\, dx~=~dt}$

5. So we want:

$\small{I = \int{\left[\frac{x^3}{x^4 + 1} \right]dx} = \int{\left[\frac{(4)x^3}{(4)(x^4 + 1)} \right]dx} = \int{\left[\frac{1}{4t} \right]dt}}$

6. This integration gives: $\small{I~=~\frac{1}{4} \log \left|t \right| ~+~\rm{C}}$

7. Substituting for t, we get:

$\small{I~=~\frac{1}{4} \log \left|x^4 + 1 \right|~+~\rm{C}}$

---

In the next section, we will see a few more miscellaneous examples.

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23.29 - Miscellaneous Examples (1) on Integrals

In the previous section, we completed a discussion on definite integrals. We saw some solved examples also. In this section, we will see some miscellaneous examples.

Solved Example 23.119
Find $\small{\int \left[\cos(6x) \sqrt{1+\sin(6x)} \right]dx}$
Solution:
1. The derivative of [1+sin(6x)] is 6cos(6x).
So we put u = 1 + sin(6x)

$\small{\Rightarrow \frac{du}{dx} = 6 \cos(6x)}$
$\small{\Rightarrow 6 \cos(6x) dx = du}$ 

2. So we want:
$\small{\int \left[\cos(6x) \sqrt{1+\sin(6x)} \right]dx~=~\int \left[\frac{6\cos(6x)}{6} \sqrt{1+\sin(6x)} \right]dx}$

$\small{~=~\int \left[\frac{1}{6} \sqrt{u} \right]dx}$

3. This integration can be done as shown below:

$\small{\int \left[\frac{1}{6} \sqrt{u} \right]dx~=~\frac{1}{6} \int \left[u^{\frac{1}{2}} \right]dx~=~\frac{1}{6} \left[\frac{u^{\frac{3}{2}}}{\frac{3}{2}} \right]+\rm{C}~=~\frac{u^{\frac{3}{2}}}{9}+\rm{C}}$

4. Substituting for u, we get:
$\small{\int \left[\cos(6x) \sqrt{1+\sin(6x)} \right]dx~=~\frac{(1+\sin(6x))^{\frac{3}{2}}}{9}+\rm{C}}$

Solved Example 23.120
Find $\small{\int \left[\frac{(x^4 - x)^{\frac{1}{4}}}{x^5} \right]dx}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{(x^4 - x)^{\frac{1}{4}}}{x^5}~=~\frac{\left[x^4 \left(1 - \frac{1}{x^3} \right) \right]^{\frac{1}{4}}}{x^5}~=~\frac{x \left[\left(1 - \frac{1}{x^3} \right) \right]^{\frac{1}{4}}}{x^5}~=~\frac{\left(1 - \frac{1}{x^3} \right)^{\frac{1}{4}}}{x^4}}$

2. The derivative of $\small{\left(1 - \frac{1}{x^3} \right)}$ is $\small{\frac{3}{x^4}}$

So we put $\small{u=1 - \frac{1}{x^3}}$

$\small{\Rightarrow \frac{du}{dx}~=~\frac{3}{x^4}~~~\Rightarrow \frac{3}{x^4}dx~=~du}$

3. So we want:

$\small{\int{\left[\frac{\left(1 - \frac{1}{x^3} \right)^{\frac{1}{4}}}{x^4} \right]dx}=\int{\left[\frac{3\left(1 - \frac{1}{x^3} \right)^{\frac{1}{4}}}{3 x^4} \right]dx}=\int{\left[\frac{\left(u \right)^{\frac{1}{4}}}{3} \right]du}}$

4. This integration can be done as shown below:

$\small{\int{\left[\frac{\left(u \right)^{\frac{1}{4}}}{3} \right]du}~=~\frac{\left(u \right)^{\frac{5}{4}}}{3(5/4)}~=~\frac{4\left(u \right)^{\frac{5}{4}}}{15}}$

4. Substituting for u, we get:

$\small{\int{\left[\frac{\left(1 - \frac{1}{x^3} \right)^{\frac{1}{4}}}{x^4} \right]dx}=\frac{4\left(1 - \frac{1}{x^3} \right)^{\frac{5}{4}}}{15}+\rm{C}=\frac{4}{15} \left(1 - \frac{1}{x^3} \right)^{\frac{5}{4}}+\rm{C}}$

Solved Example 23.121
Find $\small{\int \left[\frac{x^4}{(x-1)(x^2+1)} \right]dx}$
Solution:
1. The numerator is a polynomial of degree 4. The denominator is a polynomial of degree 3.

2. So it is not a proper rational function. We must do long division. We get:

$\small{\frac{x^4}{(x-1)(x^2+1)}\,=\,(x+1)~+~ \frac{1}{(x-1)(x^2 + 1)}}$

• The reader may write all steps involved in the long division (or any other suitable method) process.

3. In the R.H.S, the first term can be easily integrated. But the second term must be subjected to partial fraction decomposition.

• The denominator is already factorized: $\small{(x-1)(x^2 + 1)}$

   ♦ One factor is quadratic.
   
   ♦ That quadratic factor cannot be further factorized.
   
   ♦ All other factors are linear.

4. The quadratic factor $\small{(x^2+1)}$ cannot be further factorized. So this is case III. Then we are able to write:

$\small{\frac{1}{(x^2 + 1)(x - 1)}~=~\left[\frac{Ax + B}{x^2 + 1}\right]~+~\frac{A_1}{ x - 1}}$
Where A, B and A₁ are real numbers.

5. To find A, B and A₁, we make denominators same on both sides:

$\small{\frac{1}{(x^2 + 1)(x - 1)}~=~\frac{(Ax+B) (x-1)~+~A_1 (x^2+1)}{(x^2 +1)(x-1)}}$

6. Since denominators are same on both sides, we can equate the numerators. We get:

$\small{1~=~(Ax+B) (x-1)~+~A_1 (x^2+1)}$

7. After equating the numerators, we can use suitable substitution.

   ♦ Put x = 1. We get: $\small{1~=~(Ax+B) (x-1)~+~A_1 (x^2+1)}$. So A₁ = 1/2
   
   ♦ Put x = 0. We get: 1 = −B + 1/2. So B = −1/2
   
   ♦ Put x = −1. We get: 1 = (−A − (1/2))(-2) + (1/2)2. So A = −(1/2)

8. Now the result in (4) becomes:

$\small{\frac{1}{(x^2 + 1)(x - 1)}~=~\left[\frac{(-1/2)x ~-~ (1/2)}{x^2 + 1}\right]~+~\frac{1/2}{ x - 1}}$

$\small{~=~\left[\frac{-x ~-~ 1}{2(x^2 + 1)}\right]~+~\frac{1}{2(x - 1)}}$

$\small{~=~\frac{1}{2(x - 1)}~-~\frac{x}{2(x^2 + 1)}~-~\frac{1}{2(x^2 + 1)}}$

9. So based on the result in (2), we get:

$\small{\frac{x^4}{(x-1)(x^2+1)}\,=\,(x+1)~+~ \frac{1}{2(x - 1)}~-~\frac{x}{2(x^2 + 1)}~-~\frac{1}{2(x^2 + 1)}}$

10. Now the integration becomes easy. We get:

$\small{\int \left[\frac{x^4}{(x-1)(x^2+1)} \right]dx}$

$\small{~=~\frac{x^2}{2}~+~x~+~\frac{1}{2}\,\log \left|x-1 \right|~-~\frac{1}{4} \log (x^2+1)~-~\frac{1}{2} \tan^{-1}x~+~\rm{C}}$

• The reader may write all the steps involved in the integration process.

Solved Example 23.122
Find $\small{\int \left[\log(\log x)+\frac{1}{(\log x)^2} \right]dx}$
Solution:
1. Let $\small{I=\int \left[\log(\log x)+\frac{1}{(\log x)^2} \right]dx}$

$\small{~=~\int \left[\log(\log x) \right]dx~+~\int \left[\frac{1}{(\log x)^2} \right]dx~=~I_1 + I_2}$

2. First we will calculate I₁:

(a) Assigning first and second functions:

   ♦ Let first function be: $\small{f(x)=\log(\log x)}$

   ♦ Let second function be: $\small{g(x)=1}$

(b) Finding A:

$\small{A~=~\int{\left[g(x) \right]dx}~=~\int{\left[1\right]dx}~=~x}$

(c) $\small{\big[f(x) \left(A \right) \big]~=~\big[\log (\log x) \, \left(x \right) \big]}$

• This is the first term.

(d) $\small{f'(x)~=~\frac{1}{\log (x)}\left(\frac{1}{x} \right)~=~\frac{1}{x \log(x)}}$

(e) $\small{\int{\big[f'(x)\,\left(A \right)  \big]dx}~=~\int{\big[\frac{1}{x \log(x)}\,\left(x \right)\big]dx}~=~\int{\big[\frac{1}{\log(x)}\big]dx}}$

• This is the second term.

(f) So we get:

$\small{\int{\left[\log(\log x) \right]dx}~=~\text{First term - Second term}}$

$\small{~=~\big[\left(x \right) \log (\log x)\big]~-~\int{\big[\frac{1}{\log(x)}\big]dx}}$

3. In the above result, let us calculate $\small{\int{\big[\frac{1}{\log(x)}\big]dx}}$

(a) Assigning first and second functions:

   ♦ Let first function be: $\small{f(x)=\frac{1}{\log(x)}}$

   ♦ Let second function be: $\small{g(x)=1}$

(b) Finding A:

$\small{A~=~\int{\left[g(x) \right]dx}~=~\int{\left[1\right]dx}~=~x}$

(c) $\small{\big[f(x) \left(A \right) \big]~=~\big[\frac{1}{\log(x)} \, \left(x \right) \big]}$

• This is the first term.

(d) $\small{f'(x)~=~\frac{d}{dx}\left([\log(x)]^{-1} \right)~=~(-1)[\log(x)]^{-2}\left(\frac{1}{x} \right)~=~\frac{-1}{x[\log(x)]^2}}$

(e) $\small{\int{\big[f'(x)\,\left(A \right)  \big]dx}~=~\int{\big[\frac{-1}{x[\log(x)]^2}\,\left(x \right)\big]dx}~=~\int{\big[\frac{-1}{[\log(x)]^2}\big]dx}}$

• This is the second term.

(f) So we get:

$\small{\int{\left[\frac{1}{\log(x)} \right]dx}~=~\text{First term - Second term}}$

$\small{~=~\big[\frac{x}{\log(x)}\big]~-~\int{\big[\frac{-1}{[\log(x)]^2}\big]dx}}$

$\small{~=~\big[\frac{x}{\log(x)}\big]~+~\int{\big[\frac{1}{[\log(x)]^2}\big]dx}}$

4. Substituting the above result in (2), we get:

$\small{\int{\left[\log(\log x) \right]dx}~=~\big[\left(x \right) \log (\log x)\big]~-~\int{\big[\frac{1}{\log(x)}\big]dx}}$

$\small{\Rightarrow\int{\left[\log(\log x) \right]dx}~=~\big[\left(x \right) \log (\log x)\big]~-~\Bigg[\big[\frac{x}{\log(x)}\big]~+~\int{\big[\frac{1}{[\log(x)]^2}\big]dx} \Bigg]}$

$\small{\Rightarrow I_1 =\int{\left[\log(\log x) \right]dx}~=~\big[\left(x \right) \log (\log x)\big]~-~\big[\frac{x}{\log(x)}\big]~-~\int{\big[\frac{1}{(\log x)^2}\big]dx}}$

5. Now we can substitute the value of I₁ in step (1). We get:

$\small{I=I_1 + I_2}$

$\small{=\int \left[\log(\log x) \right]dx~+~\int \left[\frac{1}{(\log x)^2} \right]dx}$

$\small{=\big[\left(x \right) \log (\log x)\big]~-~\big[\frac{x}{\log(x)}\big]~-~\int{\big[\frac{1}{(\log x)^2}\big]dx}~+~\int \left[\frac{1}{(\log x)^2} \right]dx}$

$\small{=\big[\left(x \right) \log (\log x)\big]~-~\big[\frac{x}{\log(x)}\big]~+~\rm{C}}$

Solved Example 23.123
Find $\small{\int{\left[\frac{\sin(2x) \cos(2x)}{\sqrt{9 - \cos^4(2x)}} \right]dx}}$
Solution:
1.1. The derivative of $\small{\cos^2(2x)}$ is $\small{-4 \sin(2x) \cos(2x)}$.

So we put $\small{u = \cos^2(2x)}$

$\small{\Rightarrow \frac{du}{dx} = -4 \sin(2x) \cos(2x)}$
$\small{\Rightarrow -4 \sin(2x) \cos(2x) dx = du}$

2. So we want:
$\small{\int \left[\frac{(-4)\sin(2x) \cos(2x)}{(-4)\sqrt{9 - \cos^4(2x)}} \right]dx~=~\int \left[\frac{1}{(-4)\sqrt{9 - u^2}} \right]du}$

$\small{~=~\left(\frac{-1}{4} \right)\int \left[\frac{1}{\sqrt{9 - u^2}} \right]du~=~\left(\frac{-1}{4} \right)\int \left[\frac{1}{\sqrt{3^2 - u^2}} \right]du}$

3. This is a standard integral. We can write:

$\small{\left(\frac{-1}{4} \right)\int \left[\frac{1}{\sqrt{3^2 - u^2}} \right]du~=~\frac{-1}{4} \left[\sin^{-1}\frac{u}{3} \right]+\rm{C}}$

4. Substituting for u, we get:
$\small{\frac{-1}{4} \left[\sin^{-1}\left(\frac{\cos^2(2x)}{3} \right) \right]+\rm{C}}$

Solved Example 23.124
Evaluate $\small{\int_{-1}^{\frac{3}{2}}{\left[\left|x \sin(\pi x) \right| \right]dx}}$
Solution:
1. Let us determine the intervals where $\small{x \sin(\pi x)}$ is +ve or -ve.

The given interval is [−1,1.5]. This interval can be split into three: [−1,0], [0,1] and [1,1.5]

2. Consider the interval [−1,0]. In this interval, all x values are −ve. So $\small{(\pi x)}$ is −ve. It corresponds to III and IV quadrants, where sine is −ve. So $\small{x \sin(\pi x)}$ is +ve.

So for this interval, we can write:

$\small{\left|x \sin(\pi x) \right|=x \sin(\pi x)~\text{if}~-1 < x < 0}$

3. Consider the interval [0,1]. In this interval, all x values are +ve. So $\small{(\pi x)}$ is +ve. It corresponds to I and II quadrants, where sine is +ve. So $\small{x \sin(\pi x)}$ is +ve.

So for this interval, we can write:

$\small{\left|x \sin(\pi x) \right|=x \sin(\pi x)~\text{if}~0 < x < 1}$

• (2) and (3) give the same result. Also, they are adjacent intervals. So those two intervals can be combined.

4. Consider the interval [1,1.5]. In this interval, all x values are +ve. So $\small{(\pi x)}$ is +ve. It corresponds to III quadrant, where sine is −ve. So $\small{x \sin(\pi x)}$ is −ve.

So for this interval, we can write:

$\small{\left|x \sin(\pi x) \right|=-x \sin(\pi x)~\text{if}~1 < x < 1.5}$

5. Consider the exact points x = −1, x = 0, x = 1 and x = 1.5

(a) When x = −1, $\small{x \sin(\pi x)} = (-1)\sin(-\pi)=0$

So for this point, we can write:

$\small{\left|x \sin(\pi x) \right|= \pm x \sin(\pi x)~\text{if}~ x = -1}$

(b) When x = 0, $\small{x \sin(\pi x)} = (0)\sin(0)=0$

So for this point, we can write:

$\small{\left|x \sin(\pi x) \right|= \pm x \sin(\pi x)~\text{if}~ x = 0}$

(c) When x = 1, $\small{x \sin(\pi x)} = (1)\sin(\pi)=0$

So for this point, we can write:

$\small{\left|x \sin(\pi x) \right|= \pm x \sin(\pi x)~\text{if}~ x = 1}$

(d) When x = 1.5, $\small{x \sin(\pi x)} = (1.5)\sin((1.5)\pi)=-1.5$

So for this point, we can write:

$\small{\left|x \sin(\pi x) \right|= -x \sin(\pi x)~\text{if}~ x = 1.5}$

6. Combining (2), (3), (4) and (5), we can write:

$\left|x \sin(\pi x) \right| = \begin{cases} x \sin(\pi x),  & \text{if}~-1 \le x \le 1  \\[1.5ex] -x \sin(\pi x), & \text{if}~~1 \le x \le 1.5  \end{cases}$

7. The integrand changes at 1. So we will split the interval at that point. Then by applying P₂, it can be written as:

$\small{\int_{-1}^{1.5}{\left[\left|x \sin(\pi x) \right| \right]dx}}$

$\small{\,=\,\int_{-1}^{1}{\left[\left|x \sin(\pi x) \right| \right]dx}\,+\,\int_{1}^{1.5}{\left[\left|x \sin(\pi x) \right| \right]dx}}$

• We will denote it as:

$\small{\int_{-1}^{1.5}{\left[\left|x \sin(\pi x) \right| \right]dx}\,=\,I_1\,+\,I_2}$

8. Choosing the appropriate segments:

• For I₁, we must use the segment: $\small{\left|x \sin(\pi x) \right|\,=\,x \sin(\pi x),~\text{if}~-1 \le x \le 1}$

• For I₂, we must use the segment: $\small{\left|x \sin(\pi x) \right|\,=\,-x \sin(\pi x),~\text{if}~1 \le x \le 1.5}$

9. So from (7), we get:

$\small{\int_{-1}^{1.5}{\left[\left|x \sin(\pi x) \right| \right]dx}=I_1 + I_2}$

$\small{\,=\,\int_{-1}^{1}{\left[x \sin(\pi x) \right]dx}\,+\,\int_{1}^{1.5}{\left[x \sin(\pi x) \right]dx}}$

$\small{\,=\,\int_{-1}^{1}{\left[x \sin(\pi x) \right]dx}\,-\,\int_{1}^{1.5}{\left[x \sin(\pi x) \right]dx}~=~I_3 ~-~ I_4}$

10. So our next task is to find the indefinite integral: $\small{\int{\left[x \sin(\pi x) \right]dx}}$   

(a) Assigning first and second functions:

   ♦ Let first function be: $\small{f(x)= x}$

   ♦ Let second function be: $\small{g(x)=\sin(\pi x)}$

(b) Finding A:

$\small{A~=~\int{\left[g(x) \right]dx}~=~\int{\left[\sin(\pi x)\right]dx}~=~\frac{-\cos(\pi x)}{\pi}}$

(c) $\small{\big[f(x) \left(A \right) \big]~=~\big[x \, \left(\frac{-\cos(\pi x)}{\pi} \right) \big]~=~\big[\frac{-x \cos(\pi x)}{\pi}  \big]}$

• This is the first term.

(d) $\small{f'(x)~=~1}$

(e) $\small{\int{\big[f'(x)\,\left(A \right)  \big]dx}~=~\int{\big[(1)\,\left(\frac{-\cos(\pi x)}{\pi} \right)\big]dx}~=~\frac{-1}{\pi}\int{\big[\cos (\pi x)\big]dx}}$

$\small{~=~\frac{-1}{\pi} \left(\frac{\sin(\pi x)}{\pi} \right)~=~\frac{-\sin(\pi x)}{\pi^2}}$

• This is the second term.

(f) So we get:

$\small{\int{\left[x \sin(\pi x) \right]dx}~=~\text{First term - Second term}}$

$\small{~=~\big[\frac{-x \cos(\pi x)}{\pi}\big]~+~\big[\frac{\sin(\pi x)}{\pi^2}\big]}$

11. Now we can calculate I₃

$\small{\int_{-1}^{1}{\left[x \sin(\pi x) \right]dx}~=~\big[\frac{-x \cos(\pi x)}{\pi}~+~\frac{\sin(\pi x)}{\pi^2}\big]_{-1}^1}$

$\small{~=~\big[\left(\frac{-(1) \cos(\pi)}{\pi}~+~\frac{\sin(\pi)}{\pi^2} \right)~-~\left(\frac{-(-1) \cos(-\pi)}{\pi}~+~\frac{\sin(-\pi)}{\pi^2} \right)\big]}$

$\small{~=~\big[\left(\frac{-(1) (-1)}{\pi}~+~\frac{0}{\pi^2} \right)~-~\left(\frac{-(-1) (-1)}{\pi}~+~\frac{0}{\pi^2} \right)\big]}$

$\small{~=~\big[\left(\frac{1}{\pi} \right)~-~\left(\frac{(-1)}{\pi} \right)\big]~=~\frac{2}{\pi}}$

12. Next we can calculate I₄

$\small{\int_{1}^{1.5}{\left[x \sin(\pi x) \right]dx}~=~\big[\frac{-x \cos(\pi x)}{\pi}~+~\frac{\sin(\pi x)}{\pi^2}\big]_{1}^{1.5}}$

$\small{~=~\big[\left(\frac{-(1.5) \cos(1.5\pi)}{\pi}~+~\frac{\sin(1.5\pi)}{\pi^2} \right)~-~\left(\frac{-(1) \cos(\pi)}{\pi}~+~\frac{\sin(\pi)}{\pi^2} \right)\big]}$

$\small{~=~\big[\left(\frac{0}{\pi}~+~\frac{-1}{\pi^2} \right)~-~\left(\frac{-(1) (-1)}{\pi}~+~\frac{0}{\pi^2} \right)\big]}$

$\small{~=~\big[\left(\frac{-1}{\pi^2} \right)~-~\left(\frac{1}{\pi} \right)\big]~=~\frac{-1}{\pi^2}~-~\frac{1}{\pi}}$

13. Thus we get: I₃ − I₄

$\small{~=~\frac{2}{\pi}~-~\left(\frac{-1}{\pi^2}~-~\frac{1}{\pi} \right)}$

$\small{~=~\frac{2}{\pi}~+~\frac{1}{\pi^2}~+~\frac{1}{\pi}}$

$\small{~=~\frac{3}{\pi}~+~\frac{1}{\pi^2}}$

Solved Example 23.125
Evaluate $\small{\int_{1}^{4}{\left[\left|x-1 \right| ~+~\left|x-2 \right|~+~\left|x-3 \right| \right]dx}}$
Solution:
1. We can write:

$\small{\int_{1}^{4}{\left[\left|x-1 \right| ~+~\left|x-2 \right|~+~\left|x-3 \right| \right]dx}}$

= $\small{\int_{1}^{4}{\left[\left|x-1 \right|  \right]dx}~+~\int_{1}^{4}{\left[\left|x-2 \right| \right]dx}~+~\int_{1}^{4}{\left[\left|x-3 \right| \right]dx}}$

= I₁ + I₂ + I₃

2. First we will evaluate I₁:

(a) Let us determine the intervals where (x−1) is +ve or −ve. For that, first we solve the inequality: x−1<0

We have: $\small{x-1 < 0}$

$\small{\Rightarrow x < 1}$

• So when x is less than 1, (x−1) will be −ve.

• That means, when x is less than 1, $\small{\left|x-1 \right|\,=\,-(x-1)}$

(b) Next we solve the inequality:x−1>0

We have: $\small{x-1 > 0}$

$\small{\Rightarrow x > 1}$

• So when x is greater than 1, (x-1) will be +ve.

• That means, when x is greater  than 1, $\small{\left|x-1 \right|\,=\,x-1}$

(c) Also, we must solve the equation x-1 = 0

This gives x = 1

• So when x is equal to 1, (x-1) will be zero.

• That means, when x is equal to 1, $\small{\left|x-1 \right|\,=\,\pm(x-1)}$

(d) Now we can write a piece wise function:

$\left|x-1 \right| = \begin{cases} x-1,  & \text{if}~x \ge 1 \\[1.5ex] -(x-1), & \text{if}~x<1  \end{cases}$

(e) The given interval is [1,4]. So x is always greater than or equal to 1. Therefore, we need to consider the first segment only.

We can write:

$\small{I_1~=~\int_{1}^{4}{\left[\left|x-1 \right| \right]dx}\,=\,\int_{1}^{4}{\left[x-1 \right]dx}}$

$\small{\,=\,\left[\frac{x^2}{2}\,-\,x \right]_{1}^{4}}$

$\small{\,=\,\left[\frac{16}{2}\,-\,4\,-\,\left(\frac{1}{2}\,-\,1 \right) \right]}$

$\small{\,=\,\left[4\,-\,\left(\frac{-1}{2} \right) \right]~=~\frac{9}{2}}$

3. Next we will evaluate I₂:

(a) Let us determine the intervals where (x−2) is +ve or −ve. For that, first we solve the inequality: x−2<0

We have: $\small{x-2 < 0}$

$\small{\Rightarrow x < 2}$

• So when x is less than 1, (x−2) will be −ve.

• That means, when x is less than 2, $\small{\left|x-2 \right|\,=\,-(x-1)}$

(b) Next we solve the inequality:x−2>0

We have: $\small{x-2 > 0}$

$\small{\Rightarrow x > 2}$

• So when x is greater than 2, (x-2) will be +ve.

• That means, when x is greater  than 2, $\small{\left|x-2 \right|\,=\,x-2}$

(c) Also, we must solve the equation x-2 = 0

This gives x = 2

• So when x is equal to 2, (x-2) will be zero.

• That means, when x is equal to 2, $\small{\left|x-2 \right|\,=\,\pm(x-2)}$

(d) Now we can write a piece wise function:

$\left|x-2 \right| = \begin{cases} x-2,  & \text{if}~x \ge 2 \\[1.5ex] -(x-2), & \text{if}~x<2  \end{cases}$

(e) The integrand changes at 2. So we will split the interval at 2. Then by applying P₂, it can be written as:

$\small{\int_{1}^{4}{\left[\left|x-2 \right| \right]dx}\,=\,\int_{1}^{2}{\left[\left|x-2 \right| \right]dx}\,+\,\int_{2}^{4}{\left[\left|x-2 \right| \right]dx}}$

• We will denote it as:

$\small{\int_{1}^{4}{\left[\left|x-2 \right| \right]dx}\,=\,I_4\,+\,I_5}$

(f) Choosing the appropriate segments:

• For I₄, we must use the segment: $\small{\left|x-2 \right|\,=\,-(x-2),~\text{if}~x<2}$

• For I₅, we must use the segment: $\small{\left|x-2 \right|\,=\,x-2,~\text{if}~x \ge 2}$

(g) So from (e), we get:

$\small{\int_{1}^{4}{\left[\left|x-2 \right| \right]dx}\,=\,\int_{1}^{2}{\left[-(x-2) \right]dx}\,+\,\int_{2}^{4}{\left[x-2 \right]dx}}$

$\small{\,=\,(-1)\int_{1}^{2}{\left[x-2 \right]dx}\,+\,\int_{2}^{4}{\left[x-2 \right]dx}}$

$\small{\,=\,(-1)\left[\frac{x^2}{2}\,-\,2x \right]_{1}^{2}\,+\,\left[\frac{x^2}{2}\,-\,2x \right]_{2}^{4}}$

$\small{\,=\,(-1)\left[\frac{2^2}{2}\,-\,4\,-\,\left(\frac{1^2}{2}\,-\,2 \right) \right]\,+\,\left[\frac{4^2}{2}\,-\,8\,-\,\left(\frac{2^2}{2}\,-\,4 \right) \right]}$

$\small{\,=\,(-1)\left[-2\,-\,\left(\frac{-3}{2} \right) \right]\,+\,\left[0\,-\,\left(-2 \right) \right]}$

$\small{\,=\,(-1)\left[\frac{-1}{2}  \right]\,+\,\left[2  \right]~=~\frac{5}{2}}$

4. In a similar way, we can calculate I₃. We get:

$\small{\int_{1}^{4}{\left[\left|x-3 \right| \right]dx}~=~\frac{5}{2}}$ 

5. Therefore, we can write:

$\small{I = I_1 + I_2 + I_3 ~=~\frac{9}{2} + \frac{5}{2} + \frac{5}{2}~=~\frac{19}{2}}$

Solved Example 23.126
Evaluate $\small{\int_{0}^{\pi}{\left[\frac{x}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}}$
Solution:
1. Applying P₄, we can write:

$\small{I=\int_{0}^{\pi}{\left[\frac{x}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}=\int_{0}^{\pi}{\left[\frac{\pi - x}{a^2 \cos^2 (\pi - x)~+~b^2 \sin^2 (\pi - x)} \right]dx}}$

$\small{=\int_{0}^{\pi}{\left[\frac{\pi-x}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}}$

$\small{=\int_{0}^{\pi}{\left[\frac{\pi}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}~-~\int_{0}^{\pi}{\left[\frac{x}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}}$

$\small{=\int_{0}^{\pi}{\left[\frac{\pi}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}~-~I}$

2. So we can write: $\small{2I=\int_{0}^{\pi}{\left[\frac{\pi}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}}$

$\small{\Rightarrow I=\frac{\pi}{2} \int_{0}^{\pi}{\left[\frac{1}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}}$

3. Here we can apply P₆. The reader may write how P₆ is applicable.

We get: $\small{I=\frac{\pi}{2} (2) \int_{0}^{\frac{\pi}{2}}{\left[\frac{1}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}=\pi \int_{0}^{\frac{\pi}{2}}{\left[\frac{1}{a^2 \cos^2 x~+~b^2 \sin^2 x} \right]dx}}$

4. Dividing both numerator and denominator by $\small{\cos^2 x}$, we get:

$\small{I=\pi \int_{0}^{\frac{\pi}{2}}{\left[\frac{\sec^2 x}{a^2~+~b^2 \tan^2 x} \right]dx}}$

• Put $\small{u=b \tan x \Rightarrow \frac{du}{dx} = b \sec^2 x \Rightarrow b \sec^2 x dx = du}$

   ♦ Also, when x approach zero, u approach zero

   ♦ And when x approach $\small{\frac{\pi}{2}}$, u approach $\small{\infty}$

• So we want to evaluate:

$\small{I= \frac{\pi}{b} \int_{0}^{\frac{\pi}{2}}{\left[\frac{b \sec^2 x}{a^2~+~b^2 \tan^2 x} \right]dx} = \frac{\pi}{b} \int_{0}^{\infty}{\left[\frac{1}{a^2~+~u^2} \right]du}}$

5. This is a standard integral. We get:

$\small{I= \frac{\pi}{b} \left[\frac{1}{a} \tan^{-1} \frac{u}{a} \right]_0^{\infty} = \frac{\pi}{ab} \left[ \frac{\pi}{2}~-~0 \right]~=~\frac{\pi^2}{2ab}}$

Solved Example 23.127
Integrate $\small{\frac{1}{x - x^3}}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{1}{x - x^3}~=~\frac{1}{(x)\frac{x^3}{x^3} ~-~ (x^3)\frac{x^3}{x^3}}~=~\frac{1}{x^3 \left[(x)\frac{1}{x^3} ~-~ (x^3)\frac{1}{x^3} \right]}~=~\frac{1}{x^3 \left[\frac{1}{x^2} ~-~ 1 \right]}}$

2. The derivative of $\small{\left(\frac{1}{x^2} - 1 \right)}$ is $\small{\frac{-2}{x^3}}$

So we put $\small{u=\frac{1}{x^2} - 1}$

$\small{\Rightarrow \frac{du}{dx}~=~\frac{-2}{x^3}~~~\Rightarrow \frac{-2}{x^3}dx~=~du}$

3. So we want: $\small{I = \int{\left[\frac{1}{x - x^3} \right]dx}}$

$\small{=\int{\left[\frac{1}{x^3 \left[\frac{1}{x^2} ~-~ 1 \right]} \right]dx}=\int{\left[\frac{(-2)}{(-2)x^3 \left[\frac{1}{x^2} ~-~ 1 \right]} \right]dx}=\int{\left[\frac{1}{(-2) \left[u \right]} \right]du}}$

$\small{= \frac{-1}{2} \int{\left[\frac{1}{u} \right]du}~=~\frac{-\log u}{2}+\rm{C}~=~\frac{1}{2} \log \left(\frac{1}{u} \right)+\rm{C}}$

4. We wrote: $\small{u=\frac{1}{x^2} - 1~=~\frac{1 - x^2}{x^2}}$

• So $\small{\frac{1}{u}~=~\frac{x^2}{1 - x^2}}$

• Substituting this in (3), we get:

$\small{I = \int{\left[\frac{1}{x - x^3} \right]dx}~=~\frac{1}{2} \log \left|\frac{x^2}{1 - x^2} \right| +\rm{C}}$

Solved Example 23.128
Integrate $\small{\frac{1}{x \sqrt{ax - x^2}}}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{1}{x \sqrt{ax - x^2}}
~=~\frac{\left(ax - x^2 \right)^{-(1/2)}}{x}
~=~\frac{\left[ax \left(\frac{x^2}{x^2} \right)- x^2\left(\frac{x^2}{x^2} \right) \right]^{-(1/2)}}{x}
}$

$\small{
~=~\frac{(x^2)^{-(1/2)} \left[ax \left(\frac{1}{x^2} \right)- x^2\left(\frac{1}{x^2} \right) \right]^{-(1/2)}}{x}
~=~\frac{x^{-1} \left[\frac{a}{x}-1   \right]^{-(1/2)}}{x}
}$

$\small{
~=~\frac{\left[\frac{a}{x}-1  \right]^{-(1/2)}}{x^2}
}$

2. Put $\small{u = \frac{a}{x}-1}$. Then we get:

$\small{\frac{du}{dx}=\frac{-a}{x^2}}$

$\small{\Rightarrow \frac{-a}{x^2} dx~=~du}$

3. So we want: $\small{I = \int{\left[\frac{\left[\frac{a}{x}-1  \right]^{-(1/2)}}{x^2} \right]dx}= \int{\left[\frac{(-a)\left[\frac{a}{x}-1  \right]^{-(1/2)}}{(-a)x^2} \right]dx}}$

$\small{=\int{\left[\frac{\left[u  \right]^{-(1/2)}}{(-a)} \right]du}=\left(\frac{-1}{a} \right)\int{\left[u^{-(1/2)} \right]du}}$

4. This integration gives:

$\small{\left(\frac{-1}{a} \right)\left[\frac{u^{1/2}}{1/2} ~+~\rm{C_1}\right]=\left[\frac{2 u^{1/2}}{(-a)} ~+~\rm{C}\right] =\frac{-2 u^{1/2}}{a}~+~\rm{C}}$

5. Substituting for u, we get:

$\small{I= \frac{-2 }{a} \left(\frac{a}{x}-1 \right)^{1/2}~+~\rm{C}}$

Solved Example 23.129
Integrate $\small{\frac{1}{x^2(x^4 + 1)^{3/4}}}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{1}{x^2(x^4 + 1)^{3/4}}
~=~\frac{\left(x^4 + 1 \right)^{-(3/4)}}{x^2}
~=~\frac{\left[x^4 \left(\frac{x^4}{x^4} \right)+ 1\left(\frac{x^4}{x^4} \right) \right]^{-(3/4)}}{x^2}
}$

$\small{
~=~\frac{(x^4)^{-(3/4)} \left[x^4 \left(\frac{1}{x^4} \right)+ 1\left(\frac{1}{x^4} \right) \right]^{-(3/4)}}{x^2}
~=~\frac{x^{-3} \left[1+ \frac{1}{x^4}  \right]^{-(3/4)}}{x^2}
}$

$\small{
~=~\frac{\left[1+ \frac{1}{x^4}  \right]^{-(3/4)}}{x^5}
}$

2. Put $\small{u = 1+ \frac{1}{x^4}}$. Then we get:

$\small{\frac{du}{dx}=\frac{-4}{x^5}}$

$\small{\Rightarrow \frac{-4}{x^5} dx~=~du}$

3. So we want: $\small{I = \int{\left[\frac{\left[1+ \frac{1}{x^4}  \right]^{-(3/4)}}{x^5} \right]dx}= \int{\left[\frac{(-4)\left[1+ \frac{1}{x^4}  \right]^{-(3/4)}}{(-4)x^5} \right]dx}}$

$\small{=\int{\left[\frac{\left[u  \right]^{-(3/4)}}{(-4)} \right]du}=\left(\frac{-1}{4} \right)\int{\left[u^{-(3/4)} \right]du}}$

4. This integration gives:

$\small{\left(\frac{-1}{4} \right)\left[\frac{u^{1/4}}{1/4} ~+~\rm{C_1}\right]=\left[\frac{u^{1/4}}{(-1)} ~+~\rm{C}\right] =(-1)u^{1/4}~+~\rm{C}}$

5. Substituting for u, we get:

$\small{I= (-1)\left(1+ \frac{1}{x^4} \right)^{1/4}~+~\rm{C}}$

Solved Example 23.130
Integrate $\small{\frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}}}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{1}{x^{1/2} + x^{1/3}}
~=~\frac{1}{(x^{1/2})\frac{x^{1/3}}{x^{1/3}} ~+~ (x^{1/3})\frac{x^{1/3}}{x^{1/3}}}
}$

$\small{
~=~\frac{1}{x^{1/3} \left[(x^{1/2})\frac{1}{x^{1/3}} ~+~ (x^{1/3})\frac{1}{x^{1/3}} \right]}
~=~\frac{1}{x^{1/3} \left[x^{1/6} ~+~ 1 \right]}}$

2. Put $\small{u = x^{1/6}}$. Then we get:

$\small{\frac{du}{dx}=\frac{1}{6} x^{-5/6}=\frac{1}{6\, x^{5/6}}}$

$\small{\Rightarrow dx = 6\, x^{5/6} du}$

• Also, $\small{x = u^6}$, $\small{x^{1/3} = u^2}$ and $\small{x^{5/6} = u^5}$

3. So we want: $\small{I = \int{\left[\frac{1}{x^{1/2} + x^{1/3}} \right]dx}}$

$\small{=\int{\left[\frac{1}{x^{1/3} \left[x^{1/6} ~+~ 1 \right]} \right]dx}=\int{\left[\frac{6\, x^{5/6}}{u^{2} \left[u ~+~ 1 \right]} \right]du}}$

$\small{=\int{\left[\frac{6\, u^{5}}{u^{2} \left[u ~+~ 1 \right]} \right]du}=\int{\left[\frac{6\, u^{3}}{ u ~+~ 1 } \right]du}=6 \int{\left[\frac{u^{3}}{ u ~+~ 1 } \right]du}}$

4. So our next task is to integrate: $\small{\frac{u^{3}}{ u ~+~ 1 }}$

(a) The numerator is a polynomial of degree 3. The denominator is a polynomial of degree 1.

(b) So it is not a proper rational function. We must do long division. We get:

$\small{\frac{u^3}{u+1}\,=\,(u^2 - u +1)~+~ \frac{1}{(u + 1)}}$

• The reader may write all steps involved in the long division (or any other suitable method) process.

5. So we can write:

$\small{I~=~6 \int{\left[\frac{u^{3}}{ u ~+~ 1 } \right]du}=6 \int{\left[u^2 - u +1~+~ \frac{1}{u + 1} \right]du}}$

$\small{=6 \left[\frac{u^3}{3} - \frac{u^2}{2} + u ~+~ \log \left|u+1 \right| ~+~\rm{C_1}\right]}$

$\small{= 2 u^3 - 3 u^2 + 6u ~+~ 6\log \left|u+1 \right| ~+~\rm{6C_1}}$

6. Substituting for u, we get:

$\small{I= 2 \left(x^{1/6} \right)^3 - 3 \left(x^{1/6} \right)^2 + 6\left(x^{1/6} \right) ~+~ 6\log \left|\left(x^{1/6} \right)+1 \right| ~+~\rm{6C_1}}$

$\small{= 2 x^{1/2}  - 3 x^{1/3}  + 6x^{1/6}  ~+~ 6\log \left|x^{1/6}+1 \right| ~+~\rm{C}}$

Solved Example 23.131
Integrate $\small{\frac{x^3}{\sqrt{1 - x^8}}}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{x^3}{\sqrt{1 - x^8}}
~=~\frac{x^3}{\sqrt{1 - (x^4)^2}}}$

2. Put $\small{u = x^4}$. Then we get:

$\small{\frac{du}{dx}=4x^3}$

$\small{\Rightarrow 4x^3 dx~=~du}$

3. So we want: $\small{I = \int{\left[\frac{x^3}{\sqrt{1 - (x^4)^2}}\right]dx}= \int{\left[\frac{4 x^3}{4 \sqrt{1 - (x^4)^2}} \right]dx}}$

$\small{=\int{\left[\frac{1}{4 \sqrt{1-u^2}} \right]du}=\left(\frac{1}{4} \right)\int{\left[\frac{1}{\sqrt{1-u^2}} \right]du}}$

4. This is a standard integration. It gives:

$\small{\left(\frac{1}{4} \right)\left[\sin^{-1}u ~+~\rm{C_1}\right]=\frac{\sin^{-1}u}{4} ~+~\rm{C}}$

5. Substituting for u, we get:

$\small{I= \frac{\sin^{-1}(x^4)}{4}~+~\rm{C}}$

Solved Example 23.132
Integrate $\small{\frac{\cos x}{\sqrt{4 - \sin^2(x)}}}$
Solution:
1. First we will rearrange the given expression:
$\small{\frac{\cos x}{\sqrt{4 - \sin^2(x)}}
~=~\frac{\cos x}{\sqrt{2^2 - \sin^2(x)}}}$

2. Put $\small{u = \sin x}$. Then we get:

$\small{\frac{du}{dx}=\cos x}$

$\small{\Rightarrow \cos x\, dx~=~du}$

3. So we want: $\small{I = \int{\left[\frac{\cos x}{\sqrt{2^2 - \sin^2(x)}}\right]dx}= \int{\left[\frac{1}{\sqrt{2^2 - u^2}} \right]du}}$

4. This is a standard integration. It gives:

$\small{\sin^{-1}\frac{u}{2} ~+~\rm{C}}$

5. Substituting for u, we get:

$\small{I= \sin^{-1} \left(\frac{\sin x}{2} \right)~+~\rm{C}}$

---

In the next section, we will see a few more miscellaneous examples.

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