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# Methods of Integration

Examples showing how various functions can be integrated
View other versions (3)

## Introduction

The following methods of Integration cover all the Normal Requirements of A.P.; A. level; The International Baccalaureate as well as Engineering Degree Courses.

It does not cover approximate methods such as The Trapezoidal Rule or Simpson's Rule. These will be covered in another paper.

## Simple Algebraic Equations

Except when n = -1 Then

## Rational Algebraic Functions Whose Denominator Factorizes

Here is a worked example

## Rational Algebraic Functions Whose Denominators Do Not Factorize

Here are some examples

Example:
Problem

Workings

Solution


Example:
Problem

Workings


Solution
Other forms




## An Irrational Function Of The Following Type

thus the original equation can now be rewritten as :-

Example:
##### Example -
Problem

Find the integral of


Workings
let U =




The integral can now be written as :-





Solution


## An Irrational Function Containing

substitute
So the integral is now rational in
Example:
##### Example -
Problem
Find the integral of

Workings
Substitute

i.e.


Therefore


### Todo

Review the following workings

thus the integral can be written as:-




Solution
Therefore


## Using Trigonometrical Formula

Example:
##### Example -
Problem

To find the integral of


Workings
But

from which it can be shown that

Solution


### Any Trigonometrical Formula

To integrate any trigonometrical function such as
Example:
Problem
Workings


Solution
Therefore

Then
Example:
Problem
Workings

Solution


Example:
Problem

Workings



Solution


## Integration By Parts

this can also be written as:-
Example:
Problem

Workings
Solution


Example:
Problem


Workings





Solution
Therefore