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.

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 $\frac{d}{dx}(\frac{x^{3}}{3}+C)$ = $x^{2}$
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 $g$.
$\int [f(x)\pm g(x)]dx$ = $\int [f(x) dx \pm \int g(x)]dx$
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. $\int$ $\frac{dx}{x\sqrt{x^{2}-1}}$ = $\sec^{-1}|x|+c$; $|x|>1$

  • $\int 0\ dx$ = $c$
  • $\int 1\ dx$ = $x+c$
  • $\int k\ dx$ = $kx + c$
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

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