Integrating Powers of x (DP IB Applications & Interpretation (AI)): Revision Note

Integrating Powers of x

How do I integrate powers of x? 

• Powers of  are integrated according to the following formula:

  • If  then  where  and  is the constant of integration

• This is given in the formula booklet

• If the power of is  multiplied by a constant then the integral is also multiplied by that constant

  • If  then  where ,   is a constant and  is the constant of integration

•  notation can still be used with integration

• Note that the formulae above do not apply when  as this would lead to division by zero

• Don’t forget the special case:

  • 

    • e.g.  

  • This allows constant terms to be integrated

• Functions involving fractions with denominators in terms of  will need to be rewritten as negative powers of  first

  • e.g.  If  then rewrite as  and integrate

How do I integrate sums and differences of powers of x?

• The formulae for integrating powers of  apply to all integers so it is possible to integrate any expression that is a sum or difference of powers of 

  • e.g.  If  then 
    

• Products and quotients cannot be integrated in this way so would need expanding/simplifying first

  • e.g. If  then