In the previous section, we saw some standard integrals of trigonometric functions. We saw some solved examples also. In this section, we will see a few more solved examples.
Solved example 23.7
Find the following integrals:
(i) $\small{\int{\left[(4x+2) \sqrt{x^2 + x + 1} \right]dx}}$
(ii) $\small{\int{\left[\frac{1}{1\,-\, \tan x} \right]dx}}$
(iii) $\small{\int{\left[\frac{1}{1\,+\, \cot x} \right]dx}}$
(iv) $\small{\int{\left[\frac{1}{x\,+\, x \log x} \right]dx}}$
Solution:
Part (i):
1.
The derivative of
(x2+x+1) is 2x+1.
• So we put u = x2+x+1
⇒ $\small{\frac{du}{dx}~=~2x\,+\,1}$
⇒ (2x+1)dx = du
2. So we want:
$\small{\int{\left[(4x+2) \sqrt{x^2 + x + 1} \right]dx}~=~\int{\left[2 \sqrt{u} \right]du}}$
•
This integration can be done as shown below:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[2 \sqrt{u}\right] \, du}} & {~=~} &{2 \left[\frac{u^{3/2}}{3/2}~+~C_1 \right]~=~\frac{4 u^{3/2}}{3}}~+~C_2 \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\frac{4(x^2\,+\,x\,+\,1)^{3/2}}{3}~+~C} \\
\end{array}}$
• Note that, the constants C1, C2 etc., can be combined into a single constant C because, all constants, when differentiated, will give zero only.
Part (ii):
1.
The given expression can be rearranged as follows:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\frac{1}{1\,-\,\tan x}} & {~=~} &{\frac{1}{1\,-\,(\sin x / \cos x)}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\frac{\cos x}{\cos x\,-\,\sin x}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\frac{\cos x}{\sin (\pi/2 \,-\, x)\,-\,\sin x}} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{\frac{\cos x}{2 \cos(\pi/4) \sin(\pi/4\,-\, x)}} \\
{~\color{magenta} 5 } &{{}} &{{}} & {~=~} &{\frac{\cos x}{\sqrt{2} \sin(\pi/4 \,-\, x)}} \\
\end{array}}$
•
Derivative of
(π/4 − x) w.r.t x is −1.
• So we put u = (π/4 −x)
⇒ $\small{\frac{du}{dx}~=~-1}$
⇒ −dx = du
•
Also, since u = (π/4 −x), we get: x = π/4 − u
2. So we want:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[\frac{(-1)(-1)\cos x}{\sqrt{2} \sin(\pi/4 \,-\, x)}\right] \, dx}} & {~=~} &{\int{\left[\frac{-\cos (\pi/4 \,-\, u)}{\sqrt{2} \sin u}\right] \, du}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-\cos (\pi/4) \cos u ~-~\sin (\pi/4) \sin u}{\sqrt{2} \sin u}\right] \, du}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1/\sqrt 2) \cos u ~-~(1/\sqrt 2)\sin u }{\sqrt{2} \sin u}\right] \, du}} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1/\sqrt 2) \cot u~-~(1/\sqrt 2) }{\sqrt{2}}\right] \, du}} \\
{~\color{magenta} 5 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1/\sqrt 2)(\cot u ~+~1)}{\sqrt{2}}\right] \, du}} \\
{~\color{magenta} 6 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1~+~\cot u)}{2}\right] \, du}} \\
\end{array}}$
•
This integration can be done as shown below:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[\frac{-1~-~\cot u}{2}\right] \, du}} & {~=~} &{\int{\left[\frac{-1}{2}\right] \, du}~-~\int{\left[\frac{\cot u}{2}\right] \, du}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\frac{-1}{2} \int{\left[1 \right] \, du}~-~\frac{1}{2} \int{\left[\cot u \right] \, du}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\frac{-1}{2} \left[u\,+\,C_1 \right]~-~\frac{1}{2} \left[\log |\sin u |\,+\, C_2 \right]} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{-\frac{1}{2} \left[(\pi/4)\,-\,x\,+\,C_1 \right]~-~\frac{1}{2} \left[\log |\sin ((\pi/4)\,-\,x) |\,+\, C_2 \right]} \\
{~\color{magenta} 5 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,-\,(\pi/4)\,-\,C_1 \right]~+~\frac{1}{2} \left[-\log |\sin ((\pi/4)\,-\,x) |\,-\, C_2 \right]} \\
{~\color{magenta} 6 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log |\sin ((\pi/4)\,-\,x) |\,+\, C_4 \right]} \\
{~\color{magenta} 7 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|\sin (\pi/4) \, \cos x~-~\cos (\pi/4) \,\sin x \right |\,+\, C_4 \right]} \\
{~\color{magenta} 8 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|(1/\sqrt 2) \, \cos x ~-~ (1/\sqrt 2) \,\sin x \right |\,+\, C_4 \right]} \\
{~\color{magenta} 9 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|(1/\sqrt 2)(\cos x ~-~\sin x) \right |\,+\, C_4 \right]} \\
{~\color{magenta} {10} } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log (1/\sqrt 2)~-~ \log \left|(\cos x ~-~\sin x) \right |\,+\, C_4 \right]} \\
{~\color{magenta} {11} } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|(\cos x ~-~\sin x) \right |\,+\, C_5 \right]} \\
{~\color{magenta} {12} } &{{}} &{{}} & {~=~} &{\frac{x}{2}\,+\,\frac{C_3}{2}~-~\frac{\log \left|(\cos x ~-~\sin x) \right |}{2}\,+\,\frac{C_5}{2}} \\
{~\color{magenta} {13} } &{{}} &{{}} & {~=~} &{\frac{x}{2}\,-\,\frac{\log \left|(\cos x ~-~\sin x) \right |}{2}\,+\,C} \\
\end{array}}$
• Note that, the constants C1, C2, C3 etc., can be combined into a single constant C because, all constants, when differentiated, will give zero only.
Part (iii):
1.
The given expression can be rearranged as follows:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\frac{1}{1\,+\,\cot x}} & {~=~} &{\frac{1}{1\,+\,(\cos x / \sin x)}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\frac{\sin x}{\sin x\,+\,\cos x}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\frac{\sin x}{\cos (\pi/2 \,-\, x)\,+\,\cos x}} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{\frac{\sin x}{2 \cos(\pi/4) \cos(\pi/4\,-\, x)}} \\
{~\color{magenta} 5 } &{{}} &{{}} & {~=~} &{\frac{\sin x}{\sqrt{2} \cos(\pi/4 \,-\, x)}} \\
\end{array}}$
•
Derivative of
(π/4 −x) w.r.t x is −1.
• So we put u = (π/4 −x)
⇒ $\small{\frac{du}{dx}~=~-1}$
⇒ −dx = du
•
Also, since u = (π/4 −x), we get: x = π/4 − u
2. So we want:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[\frac{(-1)(-1)\sin x}{\sqrt{2} \cos(\pi/4 \,-\, x)}\right] \, dx}} & {~=~} &{\int{\left[\frac{-\sin (\pi/4 \,-\, u)}{\sqrt{2} \cos u}\right] \, du}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-\sin (\pi/4) \cos u ~+~\cos (\pi/4) \sin u}{\sqrt{2} \cos u}\right] \, du}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1/\sqrt 2) \cos u ~+~(1/\sqrt 2)\sin u }{\sqrt{2} \cos u}\right] \, du}} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1/\sqrt 2) ~+~(1/\sqrt 2) \tan u }{\sqrt{2}}\right] \, du}} \\
{~\color{magenta} 5 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1/\sqrt 2)(1~-~\tan u)}{\sqrt{2}}\right] \, du}} \\
{~\color{magenta} 6 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{-(1~-~\tan u)}{2}\right] \, du}} \\
\end{array}}$
•
This integration can be done as shown below:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[\frac{-1~+~\tan u}{2}\right] \, du}} & {~=~} &{\int{\left[\frac{-1}{2}\right] \, du}~+~\int{\left[\frac{\tan u}{2}\right] \, du}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\frac{-1}{2} \int{\left[1 \right] \, du}~+~\frac{1}{2} \int{\left[\tan u \right] \, du}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\frac{-1}{2} \left[u\,+\,C_1 \right]~+~\frac{1}{2} \left[-\log |\cos u |\,+\, C_2 \right]} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{-\frac{1}{2} \left[(\pi/4)\,-\,x\,+\,C_1 \right]~+~\frac{1}{2} \left[-\log |\cos ((\pi/4)\,-\,x) |\,+\, C_2 \right]} \\
{~\color{magenta} 5 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,-\,(\pi/4)\,-\,C_1 \right]~+~\frac{1}{2} \left[-\log |\cos ((\pi/4)\,-\,x) |\,+\, C_2 \right]} \\
{~\color{magenta} 6 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log |\cos ((\pi/4)\,-\,x) |\,+\, C_2 \right]} \\
{~\color{magenta} 7 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|\cos (\pi/4) \, \cos x~+~\sin (\pi/4) \,\sin x \right |\,+\, C_2 \right]} \\
{~\color{magenta} 8 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|(1/\sqrt 2) \, \cos x ~+~ (1/\sqrt 2) \,\sin x \right |\,+\, C_2 \right]} \\
{~\color{magenta} 9 } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|(1/\sqrt 2)(\cos x ~+~\sin x) \right |\,+\, C_2 \right]} \\
{~\color{magenta} {10} } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log (1/\sqrt 2)~-~ \log \left|(\cos x ~-~\sin x) \right |\,+\, C_2 \right]} \\
{~\color{magenta} {11} } &{{}} &{{}} & {~=~} &{\frac{1}{2} \left[x\,+\,C_3 \right]~+~\frac{1}{2} \left[-\log \left|(\cos x ~+~\sin x) \right |\,+\, C_4 \right]} \\
{~\color{magenta} {12} } &{{}} &{{}} & {~=~} &{\frac{x}{2}\,+\,\frac{C_3}{2}~-~\frac{\log \left|(\cos x ~+~\sin x) \right |}{2}\,+\,\frac{C_4}{2}} \\
{~\color{magenta} {13} } &{{}} &{{}} & {~=~} &{\frac{x}{2}\,-\,\frac{\log \left|(\cos x ~+~\sin x) \right |}{2}\,+\,C} \\
\end{array}}$
• Note that, the constants C1, C2, C3 etc., can be combined into a single constant C because, all constants, when differentiated, will give zero only.
Part (iv):
1.
The given expression can be rearranged as follows:
$\small{\frac{1}{x\,+\, x \log x}~=~\frac{1}{x(1\,+\, \log x)}}$
•
Derivative of
(1+log x) w.r.t x is: (0+1/x) = 1/x.
• So we put u = 1+ log x
⇒ $\small{\frac{du}{dx}~=~\frac{1}{x}}$
⇒ $\small{du~=~\frac{1}{x} {dx}}$
2. So we want:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[\frac{1}{x\,+\, x \log x}\right] \, dx}} & {~=~} &{\int{\left[\frac{1}{x(1\,+\, \log x)}\right] \, dx}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{1}{u}\right] \, du}} \\
\end{array}}$
•
This integration can be done as shown below:
$\small{\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\int{\left[\frac{1}{u}\right] \, du}} & {~=~} &{\int{\left[\frac{1}{x(1\,+\, \log x)}\right] \, dx}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\int{\left[\frac{1}{u}\right] \, du}} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{\log|(1\,+\, \log x)|~+~C} \\
\end{array}}$
The link below gives a few more solved examples:
Exercise 23.2
In the next section, we will see a few more solved examples.
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