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

5-2-5-edex-fm--alevel-we1-trigsub-soltn-a

(b) Find

5-2-5-edex-fm--alevel-we1-trigsub-soltn-b

 

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

5-2-5-edex-fm--alevel-we2-hypsub-soltn-a

(b) Find

5-2-5-edex-fm--alevel-we2-hypsub-soltn-b

 

 

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