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# Integration as an Inverse Process of Differentiation - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Integration as Reverse Process of Differentiation is considered one the most difficult concept.

• 44 Questions around this concept.

## Solve by difficulty

The integral value of $\int x^{4}dx$ is

The integral of $\int4^{x}dx$ is

$\text { If } \int x^{26} \cdot(x-1)^{17} \cdot(5 x-3) d x=\frac{x^{27} \cdot(x-1)^{18}}{k}+C$ where C is a constant of integration, then the value of k is equal to

$\text { The integral } \int\left(\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right) \log _2 x d x \text { is equal to }$:

If $I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x)$  dx and I (0) = 1, then $\mathrm{I}\left(\frac{\pi}{3}\right)$ is equal to :

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## Concepts Covered - 1

Integration as Reverse Process of Differentiation

Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given.

For example,

$\\\mathrm{\frac{d}{dx}(\sin x)=\cos x}\\\\\mathrm{\frac{d}{dx}\left ( x^2 \right )=2x}\\\\\mathrm{\frac{d}{dx}\left ( e^x \right )=e^x}$

In the above example, the function cos(x) is the derivative of sin(x). We say that sin(x) is an anti-derivative (or an integral) of cos(x). Similarly, x2 and ex  are the anti derivatives (or integrals) of 2x and ex respectively.

Also note that the derivative of a constant  (C) is zero. So we can write the above examples as:

$\\\mathrm{\frac{d}{dx}(\sin x+c)=\cos x}\\\\\mathrm{\frac{d}{dx}\left ( x^2+c \right )=2x}\\\\\mathrm{\frac{d}{dx}\left ( e^x +c\right )=e^x}$

Thus, anti derivatives (or integrals) of the above functions are not unique. Actually, there exist infinitely many anti-derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers.

For this reason C is referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function.

If the function F(x) is an antiderivative of f(x), then the expression F(x) + C is the indefinite integral of the function f(x) and is denoted by the symbol ∫ f(x) dx.

By definition,

$\\\mathrm{\int f(x)dx=F(x)+c,\;\;\;where\;\;F'(x)=f(x)\;\;and\;\;'c' \;is\;constant.}$

Here,

$\begin{array}{c||c} \hline\\ \mathbf { Symbols/Terms/Phrases } & \mathbf{Meaning} \\\\ \hline \hline \\\int f(x) d x & {\text { Integral of } f \text { with respect to } x} \\\\ \hline \\f(x) \text { in } \int f(x) d x & {\text { Integrand }} \\\\\hline \\ x \text { in } \int f(x) d x & {\text { Variable of integration }} \\\\\hline \\ \text{An integral of f } & {\text { A function F such that F'(x) = f(x)}}\\\\\hline \end{array}$

Rules of integration

f(x) and g(x) are functions with antiderivatives ∫ f(x) and  ∫ g(x) dx. Then,

$\\ {\text { (a) } \int \operatorname{kf}(x) d x=k \int f(x) d x \text { for any constant } k .} \\ {\text { (b) } \int(f(x)+g(x)) d x=\int f(x) d x+\int g(x) d x} \\ {\text { (c) } \int(f(x)-g(x)) d x=\int f(x) d x-\int g(x) d x}$

Standard Integration Formulae

$\\\mathrm{Since,\;\;\frac{d}{dx}\left ( F(x) \right )=f(x)\;\;\Leftrightarrow \;\;\int f(x)dx=F(x)+c}$

Based on this definition and various standard formulas (which we studied in Limit, Continuity and Differentiability) we can obtain the following important integration formulae,

$\\\mathrm{1.\;\;\frac{d}{d x}\left({kx}\right)=k \Rightarrow \int kd x=kx+C, \text{ where k is a constant}}\\\\\mathrm{2.\;\;\frac{d}{d x}\left(\frac{x^{n+1}}{n+1}\right)=x^{n}, n \neq-1 \Rightarrow \int x^{n} d x=\frac{x^{n+1}}{n+1}+C, n \neq-1}\\\\\mathrm{3.\;\;\frac{d}{d x}(\log |x|)=\frac{1}{x} \Rightarrow \int \frac{1}{x} d x=\log |x|+C, \text { when } x \neq 0}\\\\\mathrm{4.\;\;\frac{d}{d x}\left(e^{x}\right)=e^{x} \Rightarrow \int e^{x} d x=e^{x}+C}\\\\\mathrm{5.\;\;\frac{d}{d x}\left(\frac{a^{x}}{\log _{e} a}\right)=a^{x}, a>0, a \neq 1\;\;\Rightarrow \int a^{x} d x=\frac{a^{x}}{\log _{e} a}+C}$

## Study it with Videos

Integration as Reverse Process of Differentiation

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## Books

### Reference Books

#### Integration as Reverse Process of Differentiation

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

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