# Edexcel A Level Further Maths: Core Pure:复习笔记5.2.5 Integration by Substitution

### Integrating using Trigonometric Substitutions

The integrals covered in this revision note are based on the standard results

and

These are given in the formulae booklet

#### How do I know when to use a trigonometric substitution in integration?

There are three main types of problem

•  Type 1
Showing the standard results using a substitution (α may have a value)
The substitution will not be given in such cases
e.g.  Use a suitable substitution to show that
Let

The general idea in these types of problems is to reduce the denominator to a single term, often involving the identity , so it can be integrated using standard results or techniques

• Type 2
Reverse chain rule, possibly involving some factorising in the denominator and using ‘adjust’ and ‘compensate’ if necessary
e.g.  Find
• Type 3
The denominator contains a three-term quadratic expression – i.e. there is an x term
In such cases complete the square and use reverse chain rule
e.g.  Find

• (This works since , so effectively there is no reverse chain rule involved)
• A fourth type of problem may involve a given substitution but the skills to solve these are covered in the A Level Mathematics course

#### How do I use a trigonometric substitution to find integrals?

• STEP 1
Identify the type of problem and if a substitution is required
Determine the substitution if needed
• STEP 2
For Type 1 problems, differentiate and rearrange the substitution; change everything in the integral
For Type 2 problems, ‘adjust’ and ‘compensate’ as necessary
For Type 3 problems complete the square
• STEP 3
Integrate using standard techniques and results, possibly from the formulae booklet
For definite integration, a calculator may be used but look out for exact values being required, a calculator may give an approximation
• STEP 4
Substitute the original variable back in if necessary – this shouldn’t be necessary for definite integration
For indefinite integration, simplify where obvious and/or rearrange into a required format

#### Why is arccos x not involved in any of the integration results?

For integration the "-" at the start can be treated as the constant "-1" and so integrating would lead to "-arcsin ..."

• i.e.

#### Exam Tip

• The general form of the functions involving trigonometric and hyperbolic functions are very similar
• Be clear about which form needs a trigonometric substitution and which form need a hyperbolic substitution
• Always have a copy of the formula booklet to hand when practising these problems

#### Worked Example

(a) Use an appropriate substitution to show that

(b) Find

### Integrating using Hyperbolic Substitutions

The integrals covered in this revision note are based on the standard results

These are given in the formulae booklet

#### How do I know when to use a hyperbolic substitution in integration?

There are three main types of problem

• A fourth type of problem may involve a given substitution but the skills to solve these are covered in the A Level Mathematics course, although hyperbolic functions are not

#### How do I use a hyperbolic substitution to find integrals?

• STEP 1
Identify the type of problem and if a substitution is required
Determine the substitution if needed
• STEP 2
For Type 1 problems, differentiate and rearrange the substitution; change everything in the integral
For Type 2 problems, ‘adjust’ and ‘compensate’ as necessary
For Type 3 problems complete the square
• STEP 3
Integrate using standard techniques and results, possibly from the formulae booklet
For definite integration, a calculator may be used but look out for exact values being required, a calculator may give an approximation
• STEP 4
Substitute the original variable back in if necessary – this shouldn’t be necessary for definite integration
For indefinite integration, simplify where obvious and/or rearrange into a required format

#### Is artanh x involved in integration?

• The standard result, given in the formulae booklet is

with the alternative result  also given
• Problems involving these often involve partial fractions (since  is the difference of two squares) leading to the 'ln' result
• If you happen to recognise the integral and can use the formulae booklet result involving "artanh" to solve a problem, then do so!

#### Exam Tip

• The general form of the functions involving trigonometric and hyperbolic functions are very similar
• Be clear about which form needs a trigonometric substitution and which form need a hyperbolic substitution
• Always have a copy of the formula booklet to hand when practising these problems

#### Worked Example

(a) Use an appropriate substitution to show that

(b) Find