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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

## Irrational Algebraic Fraction Of The Following Kind 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

## Simple Trigonometrical Functions     ## 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:
##### Example -
Problem
\int \cosec\:x\:dx = \int \frac{1}{sin\,x}\:dx
Workings
Solution
Therefore
\int \cosec\:x\:dx = \ln\,\tan\frac{x}{2} + C

## Any Hyperbolic Function

### Simple Equations  ### Any Hyperbolic Equation  Then  Example:
##### Example -
Problem
\int \sech\:\phi\:d\phi=\int\frac{2U}{1+U^2}\:.\:\frac{1}{U}\:dU
Workings
Solution

## Integration Of Irrational Equations Of The Following Type Using Trigonometrical Substitution Example:
Problem
Workings
Solution

## Integration By Parts   this can also be written as:- Example:
Problem
Workings
Solution

Example:
Problem

Workings

Solution
Therefore