In the previous section, we completed a discussion on applications of derivatives. In this chapter, we will see Integrals.
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We began our discussion on derivatives in section 13.10.
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There we saw the equation for finding the distance traveled by a freely falling body as: s = 4.9t2.
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Since the distance s depends on time t, we can say that, s is a function of t. We write this as: s = f (t) = 4.9t2
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Derivative of this function is given by: ${\frac{ds}{dt}\,=\,f'(t) \,=\,9.8t}$
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This derivative can be used to find the instantaneous velocity at any instant t. For example, at the instant when the stop-watch shows 8 s, the velocity will be: f '(8) = 9.8(8) = 78.4 m/s.
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Now let us think in reverse:
If we are given the derivative, can we find the original function?
This chapter tries to answer the above question.
The process of integration
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Suppose that, we are given a derivative f '(x).
♦ The process of finding the original function f(x) is known as integration.
♦ This process is also known as anti differentiation.
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In fact, integration is the inverse process of differentiation.
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The original function that we obtain by integration is called primitive.
Let us see some basic details about integration. It can be written in 5 steps:
1. First we will see three examples:
Example (i): We know that ${\frac{d}{dx}(\sin x)\,=\,\cos x}$
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That is., cos x is the derivative of sin x.
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This is same as: sin x is an anti derivative of cos x
♦ An anti derivative is also called an integral.
Example (ii): We know that ${\frac{d}{dx}\left(\frac{x^3}{3} \right)\,=\,x^2}$
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That is., x2 is the derivative of ${\frac{x^3}{3}}$.
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This is same as: ${\frac{x^3}{3}}$ is an integral of x2.
Example (iii): We know that ${\frac{d}{dx}\left(e^x \right)\,=\,e^x}$
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That is., ex is the derivative of ex.
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This is same as: ex is an integral of ex.
2. Recall that, the derivative of a constant function is zero.
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Now consider the example (i) that we saw above.
The original function may be the sum of sin x and a constant function C (where C is any real number). Even then, we will get the same derivative:
${\frac{d}{dx}(\sin x\,+\,C)\,=\,\cos x \,+\, 0 \,=\, \cos x}$
So the integral is: sin x + C
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Similarly, consider the example (ii).
The original function may be the sum of ${\frac{x^3}{3}}$ and a constant function C (where C is any real number). Even then, we will get the same derivative:
${\frac{d}{dx}\left(\frac{x^3}{3} \,+\, C \right)\,=\,x^2 \,+\, 0 \,=\, x^2}$
So the integral is ${\frac{x^3}{3} \,+\, C}$
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Similarly, consider the example (iii).
The original function may be
the sum of ${e^x}$ and a constant function C (where C is
any real number). Even then, we will get the same derivative:
${\frac{d}{dx}(e^x \,+\, C )\,=\,e^x \,+\, 0 \,=\, e^x}$
So the integral is ${e^x \,+\, C}$
3. The constant C can be chosen arbitrarily from the set of real numbers. That means, C can be any random real number.
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So there are infinite possibilities for the integral. That means, there will be infinite integrals for a given derivative.
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For each of those infinite integrals, the derivative will be the same.
♦ The constant C is called arbitrary constant.
♦ The constant C is also called constant of integration.
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By varying C, we can get infinite anti derivatives (integrals) for any given derivative.
4. Since there are infinite integrals with the same derivative, we will write a general form. It can be explained in 4 steps:
(i) Suppose that, we are given a derivative.
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Recall that, derivative is also a function. So we will denote the given derivative as f(x).
(ii) Let the integral obtained by the integration of f(x), be F(x).
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Then we can write: $\frac{d}{dx} \left[F(x) \,+\, C \right]\,=\,f(x)$
♦ Where C is the constant of integration.
(iii) From the above expression, it is clear that:
By the integration of f(x), we will get F(x) + C
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This [F(x) + C] is a function.
♦ F(x) is unique.
♦ C can have infinite values.
(iv) Since C can have infinite values, there are infinite functions of the form [F(x) + C]
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We can write a set of all those functions as: {F+C, C ∈ R}
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This set is the general form of all integrals of f.
5. Now we will see an important property of integrals. It can be written in 6 step:
(i) Suppose that, we are given a function f.
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Pick any two integrals from among the infinite possible integrals. Let the two picked integrals be:
♦ F(x) + C1
♦ F(x) + C2
(ii) So we have two functions: F(x) + C1 and F(x) + C2
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Let us name them as g and h. So we have:
♦ g(x) = F(x) + C1
♦ h(x) = F(x) + C2
(iii) Now we can write the difference of g and h. We get:
g(x) − h(x) = [F(x) + C1] − [F(x) + C2] = [C1 − C2]
(iv) But [C1 − C2] is a constant. Let us call it C3.
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So we can write: g(x) − h(x) = C3
(v) Let us differentiate [g(x) − h(x)]. We get:
$\frac{d}{dx}[g(x) - h(x)]\,=\,\frac{d}{dx}[C_3]\,=\,0$
(vi) From (ii), we know that, g and h have the same derivative.
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So we can write:
If g and h are any two functions which have the same derivative, then the derivative of the "difference of g and h" will be zero.
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We have seen the basics of integration. In short, we can write:
We are given the derivative f(x). The process of finding the corresponding [F(x) + C] is called integration.
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We use the symbol $\int{f(x) \, dx}$ to denote the process of integration.
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When we see this symbol, we say:
Integration of f with respect to x.
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So we can write: $\int{f(x) \, dx}~=~F(x) \,+\,C$
Following table shows the meanings of various symbols/terms/phrases related to integration.
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| Table 23.1 |
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When we learned about derivatives, we saw many useful formulas. For example: $\frac{d}{dx} (\tan x) \,=\, \sec^2 x$.
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This formula can be used to write the corresponding formula for integral. We get: $\int{\sec^2 x \, dx}\,=\,\tan x \,+\, C$
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In this way, using the already known derivative formulas, we can make a list of integral formulas. A small list is given below:
$\begin{array}{ll} {~\color{magenta} {} } &{{}} &{\text{Derivatives}} & {{}} &{\text{Integrals}} \\
{~\color{} (i) } &{{}} &{\frac{d}{dx}\left(\frac{x^{n+1}}{n+1} \right)~=~x^n} & {\implies} &{\int{x^n \, dx}~=~ \frac{x^{n+1}}{n+1} \,+\, C} \\
{~\color{} {} } &{{}} &{{}} & {{}} &{{}} \\
{~\color{} (ii) } &{{}} &{\frac{d}{dx}\left(x \right)~=~1} & {\implies} &{\int{(1) \, dx}~=~ x \,+\, C} \\
{~\color{magenta} {} } &{{}} &{{}} & {{}} &{{}} \\
{~\color{} (iii) } &{{}} &{\frac{d}{dx}\left(\sin x \right)~=~\cos x} & {\implies} &{\int{\cos x \, dx}~=~ \sin x \,+\, C} \\
{~\color{magenta} {} } &{{}} &{{}} & {{}} &{{}} \\
{~\color{} (iv) } &{{}} &{\frac{d}{dx}\left(\cos^{-1} x \right)~=~ \frac{-1}{\sqrt{1 - x^2}}} & {\implies} &{\int{\frac{-1}{\sqrt{1 - x^2}} \, dx}~=~ \cos^{-1} x \,+\, C} \\
{~\color{magenta} {} } &{{}} &{{}} & {{}} &{{}} \\
{~\color{} (v) } &{{}} &{\frac{d}{dx}\left(\log |x| \right)~=~\frac{1}{x}} & {\implies} &{\int{\frac{1}{x} \, dx}~=~ \log |x| \,+\, C} \\
{~\color{} {} } &{{}} &{{}} & {{}} &{{}} \\
\end{array}$
• The reader may add as many items as possible, to this list. We will be using the items in the list to solve complicated problems.
In the next section, we will see geometrical interpretation of indefinite integral.
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