Integrating with Inverse Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Integrating with Inverse Trigonometric Functions

arcsin, arccos and arctan are (one-to-one) functions defined as the inverse functions of sine, cosine and tangent respectively.

What are the antiderivatives involving the inverse trigonometric functions?

• 

• 

• Note that the antiderivative involving  would arise from

  
  
  

  • However, the negative can be treated as a coefficient of -1 and so
    

    
    

  • Similarly,

• Unless a question requires otherwise, stick to the first two results

• These are listed in the formula booklet the other way round as ‘standard derivatives’

• For the antiderivative involving , note that  is the same as

How do I integrate these expressions if the denominator is not in the correct form?

• Some problems involve integrands that look very similar to the above

  • but the denominators start with a number other than one

  • there are three particular cases to consider

• The first two cases involve denominators of the form (with or without the square root!)

  • In the case  (i.e. denominator of the form ) there are two standard results

    • 

    • 

    • Both of these are given in the formula booklet

    • Note in the first result,  could be written 

  • In cases where  then the integrand can be rewritten by taking a factor of

    • the factor will be a constant that can be taken outside the integral

    • the remaining denominator will then start with 1

    • e.g. 

• The third type of problem occurs when the denominator has a (three term) quadratic

  • i.e.  denominators of the form
    (a rearrangement of this is more likely)

    • the integrand can be rewritten by completing the square

    • e.g. 
      This can then be dealt with like the second type of problem above with "" replaced by ""

    • This works since the derivative of  is the same as the derivative of 
      There is essentially no reverse chain rule to consider