Integration by Substitution (Edexcel A Level Further Maths): Revision Note

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 ( may have a value)
  The substitution will not be given in such cases
  e.g.  Use a suitable substitution to show that 
  Let , so  and 
  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. 

Integrating using Hyperbolic 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 hyperbolic substitution in integration?

There are three main types of problem

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