Integration by Substitution & by Parts (DP IB Analysis & Approaches (AA)): Revision Note

Integration by Substitution

What is integration by substitution?

• Integration by substitution is used when an integrand where reverse chain rule is either not obvious or is not spotted

  • in the latter case it is like a “back-up” method for reverse chain rule

How do I use integration by substitution?

• For instances where the substitution is not obvious it will be given in a question

  • e.g.  Find  using the substitution 

• Substitutions are usually of the form 

  • in some cases and other variations are more convenient

    • as these would not be obvious, they would be given in a question

  • if need be, this can be rearranged to find  in terms of 

• Integration by substitution then involves rewriting the integral, including “” in terms of 
  STEP 1
  Name the integral to save rewriting it later
  Identify the given substitution 
  

  STEP 2
  Find and rearrange into the form such that (some of) the integral can be rewritten in terms of 
  

  STEP 3
  If limits are involved, use to change them from  values to  values 
  

  STEP 4
  Rewrite the integral so everything is in terms of  rather than 
  This is the step when it may become apparent that  is needed in terms of 
  

  STEP 5
  Integrate with respect to u and either rewrite in terms of  or apply the limits using their  values

• For quotients the substitution usually involves the denominator

• It may be necessary to use ‘adjust and compensate’ to deal with any coefficients in the integrand

• Although can be treated like a fraction it should be appreciated that this is a ‘shortcut’ and the maths behind it is beyond the scope of the IB course

Integration by Parts

What is integration by parts?

• Integration by parts is generally used to integrate the product of two functions

  • however reverse chain rule and/or substitution should be considered first

    • e.g. can be solved using reverse chain rule or the substitution 

  • Integration by parts is essentially ‘reverse product rule’

    • whilst every product can be differentiated, not every product can be integrated (analytically)

What is the formula for integration by parts?

• 

• This is given in the formula booklet alongside its alternative form 

How do I use integration by parts?

• For a given integral  and (rather than and ) are assigned functions of 

• Generally, the function that becomes simpler when differentiated should be assigned to 

• There are various stages of integrating in this method

  • only one overall constant of integration (“+c”) is required

  • put this in at the last stage of working

  • if it is a definite integral then “+c” is not required at all 
    
    STEP 1
    Name the integral if it doesn’t have one already!
    This saves having to rewrite it several times – I is often used for this purpose.
    e.g. 
    
    STEP 2
    Assign and .
    Differentiate  to find and integrate  to find 
    e.g. 

    
    STEP 3
    Apply the integration by parts formula
    e.g. 
    
    STEP 4
    Work out the second integral, 
    Now include a “+c” (unless definite integration) 
    e.g.   
    
    STEP 5
    Simplify the answer if possible or apply the limits for definite integration
    e.g. 

• In trickier problems other rules of differentiation and integration may be needed

  • chain, product or quotient rule

  • reverse chain rule, substitution

Can integration by parts be used when there is only a single function?

• Some single functions (non-products) are awkward to integrate directly

  • e.g. , , ,

• These can be integrated using parts however

  • rewrite as the product ‘’ and choose  and 

  • 1 is easy to integrate and the functions above have standard derivatives listed in the formula booklet
    

Repeated Integration by Parts

When will I have to repeat integration by parts?

• In some problems, applying integration by parts still leaves the second integral as a product of two functions of 

  • integration by parts will need to be applied again to the second integral

• This occurs when one of the functions takes more than one derivative to become simple enough to make the second integral straightforward

  • These functions usually have the form 

How do I apply integration by parts more than once?

STEP 1
Name the integral if it doesn’t have one already!

STEP 2
Assign and .  Find and 

STEP 3
Apply the integration by parts formula

STEP 4
Repeat STEPS 2 and 3 for the second integral

STEP 5
Work out the second integral and include a “+c” if necessary

STEP 6
Simplify the answer or apply limits

What if neither function ever becomes simpler when differentiating?

• It is possible that integration by parts will end up in a seemingly endless loop

  • consider the product 

  • the derivative of  is 

    • no matter how many times a function involving  is differentiated, it will still involve

  • the derivative of  is 

    • would then have derivative , and so on

    • no matter how many times a function involving  or  is differentiated, it will still involve or 

• This loop can be trapped by spotting when the second integral becomes identical to (or a multiple of) the original integral

  • naming the original integral () at the start helps

  •  then appears twice in integration by parts

    • e.g. 
      where are parts of the integral not requiring further work

  • It is then straightforward to rearrange and solve the problem
    

    • e.g.