5.9.2 Further Techniques of Integration

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

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
    
    

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 never 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.