MAT135H1 Study Guide - Quotient Rule, Differentiation Rules, Chain Rule

To solve: The integral of 

Formulas to be used: (in order of appearance in the Solution part below)

• Trigonometric double-angle identity: 
• Difference rule of integration:   
• Constant multiple rule of integration:   
• Constant multiple rule of differentiation: 
• Trigonometric rule of integration: 

Solution:

The given function  has no direct integral, but can be solved by thinking of a trigonometric relationship that can be easily manipulated and integrated. In this case, the double-angle identity for cosine is used.  

                Using the double-angle identitiy for cosine

                Isolating the  term on the left side

             Making the coefficient of  term to be 1

The working equation for  can now be integrated using -substitution, as will be shown.

    Integrating both sides of the equation

                      Taking the integral using the difference rule  and

                                                          constant multiple rule 

 and                          Using -substitution and differentiating it using the constant multiple rule 

                  Replacing the values of  and 

                    Taking the integral using the trigonometric rule 

                  Substituting back the value of , which is 

Answer: