Integral Calculus

Integral calculus is mainly concerned with summing the values of a function over a particular range, and is particularly  useful for finding the area of an irregular shape.

Integration is the process of finding the anti-derivative.

means the indefinite integral of the function is to be calculated with respect to x.

The anti-derivative F(x) is called the integral,

c is called the constant of integration.

 

so  differentiating the integral results in the original function.

Basic Rules of Integration

 

 

Examples

 

Examples

 

 

  

 

∫(ax +b)ⁿ

 

Example

 

 

Integrating Trig Functions

Example

Integrating Logarithms and Exponentials

 

           

 

 

 

Examples

 

 

 

 

Integration by substitution

This can be used when the function to be integrated

 

Examples

 

 

 

This is very useful for for finding integrals such as sin³xdx

 

Proof

so

 

 

When finding definite integrals, remember to change the limits !

 

Sometimes it is easier to substitute dx

 

 

Common Forms

Examples

 

 

 

Examples

 

 

Example