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

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’

  • Though note that 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, parts of the integrand are assigned to be  and (rather than and )

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

• There are various stages of integrating using this method

  • However 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 to save rewriting it later ( is often used)

  • 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 differentiation to become simple enough to make the second integral straightforward

  • This often involves integrands of the form 

How do I apply integration by parts more than once?

• STEP 1
  Name the integral to save rewriting it later ( is often used)
   

• 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 is parts of the original integral that have already been integrated

  • It is then straightforward to rearrange and solve the problem

    • e.g.  

  • Don't forget to include a constant of integration ('+c') at the end