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Methods of Integration

Examples showing how various functions can be integrated
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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:
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

Any Hyperbolic Equation

Then
Example:
Example -
Problem
\int \sech\:\phi\:d\phi=\int\frac{2U}{1+U^2}\:.\:\frac{1}{U}\:dU
Workings

Solution


Example:
Problem

Workings



Solution


Integration By Parts

this can also be written as:-
Example:
Problem

Workings

Solution


Example:
Problem


Workings





Solution
Therefore