Integration is a reverse process of differentiation. Suppose you consider a function such as f(x) = 2x, then F(x) = x2 that's very simple because F'(x) = 2x. Hence the function F(X) is called the base function of f(x). So now we proceed even telling this as F(x) = x2+ C. Now if we want to find the base function then we use ∫ f(x) dx which gives x2+C. The symbol ∫ is called the indefinite integral.

## Indefinite Integral Definition

A function $F(x)$ is an antiderivative of a function $f(x)$ if

$F^{'}(x)$ = $f(x)$

for all $x$ in the domain of $f$. The set of all antiderivative of $f$ is the indefinite integral of $f$ with respect to $f$ denoted by

$\int f(x)dx$

The symbol $\int$ is an integral sign. The function $f$ is the integrand of the integral and $x$ is the variable of integration.

Once we have found one antiderivative $F$ of a function $f$, the other antiderivative of $f$ differ from $F$ by a constant. We indicate this in integral notation in the following way

$\int f(x)dx$ = $F(x) + C$

The constant C is the constant of integration or arbitrary constant. The above equation read "The definite integral of $f$ with respect to $x$ is $F(x)+C$." When we find $F(x)+C$, we say that we have integrated and evaluvated the integral.

Example: $\int x^{2}dx$ = $\frac{x^{3}}{3}$ + $C$ because $\frac{d}{dx}(\frac{x^{3}}{3}+C)$ = $x^{2}$

## Properties of Indefinite Integrals

There are certain rules for indifinite integration
• Constant multiple rule: A function ia antiderivative of a constant multiple $kf$ of a function $f$ if and only if it is $k$ times an atiderivative of $f$.
$\int k f(x) dx$ = $k\int f(x)dx$
• Rule of negatives: In paricular, a function is an antiderivative of $-f$ if and only if it is the negative of an antiderivative of $f$.
$\int – f(x)dx$ = $-\int f(x)dx$

• Sum and difference rule: A function is an antiderivative of a sum or difference $f\pm g$ if and only if it is the sum or difference of an antiderivative of $f$ and an antiderivative of $g$.
$\int [f(x)\pm g(x)]dx$ = $\int [f(x) dx \pm \int g(x)]dx$

## Table of Indefinite Integrals

The following table contain all the integrals which will help us to simplify the problems. The following results are direct consiquence of the definition of an integral.

1. $\int x^{n}dx$ = $\frac{x^{n+1}}{n+1}$ + $c$, $n \neq -1$

2. $\int$ $\frac{1}{x}$$dx = \log|x| + c 3. \int e^{x}dx = e^{x}+ c 4. \int a ^{x}dx = \frac{a^{x}}{\log_{e}a} + c 5. \int \sin x dx = -\cos x dx + c 6. \int \cos x dx = \sin x + c 7. \int \sec^{2} x dx = \tan x + c 8. \int cosec ^{2} x dx = -\cot x + c 9. \int \sec x \tan x dx = \sec x + c 10. \int cosec\ x \cot x dx = -cosec\ x + c 11. \int \tan x dx = -\log|\cos x|+ c = \log|\sec x|+ c 12. \int \cot x dx = \log|\sin x|+ c 13. \int \sec x dx = \log|\sec x + \tan x|+ c 14. \int cosec\ x dx = \log| cosec\ x - \cot x|+ c 15. \int \frac{dx}{\sqrt{1-x^{2}}} = \sin^{-1}x + c; |x|<1 16. \int \frac{dx}{1+x^{2}} = \tan^{-1}x+c 17. \int \frac{dx}{x\sqrt{x^{2}-1}} = \sec^{-1}|x|+c; |x|>1 • \int 0\ dx = c • \int 1\ dx = x+c • \int k\ dx = kx + c ## Indefinite Integral Examples Given below are some of the examples on indefinite integrals. ### Solved Examples Question 1: Evaluate \int xe^{x}dx Solution: Let \int xe^{x}dx = x\int e^{x}dx - 1\int e^{x}dx = xe^{x}- \int e^{x}dx = xe^{x}-e^{x}+c Question 2: Evaluate \int \sin^{2}xdx Solution: \int \sin^{2}xdx = \int \frac{1-\cos2x}{2}dx = \frac{1}{2}$$\int (1-\cos2x)dx$

= $\frac{1}{2}$$\int dx - \frac{1}{2}$$\int \cos2x dx$

= $\frac{1}{2}$$x- \frac{1}{2}\frac{\sin2x}{2}+ c = \frac{x}{2} - \frac{\sin2x}{4} + c Question 3: Find the general indefinite integral \int (10x^{3}-2 \sec^{2}x)dx Solution: Using the table above ,we have , \int (10x^{3}-2 \sec^{2}x)dx = 10\int x^{3}dx - 2\int \sec^{2}xdx = 10 \frac{x^{4}}{4} - 2\tan x+c = \frac{5}{2}$$x^{4}-2\tan x+ c$

Question 4: Find $\int$ $\frac{\cos x}{\sin^{2}x}dx$
Solution:

$\int$ $\frac{\cos x}{\sin^{2}x}dx$ = $\int$ $\frac{1}{\sin x} \frac{\cos x}{\sin x}dx$

= $\int \csc x \cot x dx$

= $-csc x + c$

Question 5: Evaluate the integral $\int (e^{x}+3\cos x- 4x^{3}+2)dx$
Solution:

$\int (e^{x}+3\cos x- 4x^{3}+2)dx$ = $\int e^{x}dx+3\int \cos x dx- 4\int x^{3} dx+2\int dx$

= $e^{x}+3\sin x-$$\frac{4x^{4}}{4}$+2x+c

= $e^{x}+3\sin x-x^{4}+2x+c$