Integrating Powers of x (DP IB Analysis & Approaches (AA)): Revision Note

Integrating Powers of x

How do I integrate powers of x?

• Powers of are integrated according to the following formulae:

  • 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 and 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

• Remember the special case:

  • 

    • e.g.  

  • This allows constant terms to be integrated

• Functions involving roots will need to be rewritten as fractional powers of first

  • eg. If then rewrite as and integrate

• 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    

• The formulae for integrating powers of apply to all rational numbers 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 this way so would need expanding/simplifying first

  • e.g.  If then

What might I be asked to do once I’ve found the anti-derivative (integrated)?

• With more information the constant of integration,, can be found

• The area under a curve can be found using integration