As Maths

Integration Definitions:

1.

The action or process of integrating.

2. In mathamatics:

The finding of an integral or integrals.

If that did not answer your question then look at this page for a more mathamatical explanation:

http://mathsfirst.massey.ac.nz/Calculus/integration/basics/definition.html

For powers of x, we have:

$\int{x^n}dx=\frac{x^{n+1}}{n+1}\text{ as long as }n\ne-1$   

.

It is not just divide by (power + 1) and add 1 to the power.

Examples:

$\displaystyle\int\cos x dx = \sin x + C\\ \displaystyle\int \sec^2 x dx = \tan x + C\\ \text{OR even more complicated:}\\ \displaystyle\int \sqrt[3]{\tan x}dx = -\dfrac{1}{2}\ln|1+\tan^{2/3}x|+\dfrac{1}{4}|\tan^{4/3}x-\tan^{2/3}x + 1|+\dfrac{\sqrt{3}}{2}\arctan\left(\dfrac{2\tan^{2/3}x-1}{\sqrt3}\right) + C$