The following are derived from the differentiation of the six basic functions.  

 

To generalize the formulas,

 

Example: Integrate the following with respect to x,

          (a) 3x² sec² (x³)

 

Solution:

 

(a)

          

 

 

Integration of the tangent and cotangent functions: Change the function into a quotient of sines and cosines, then apply integration to change it into a logarithmic function.

                                                            

Work out the integration of cotangent yourself as practice. The result is ln |sinx| + c.

 

Sometimes, the trigonometric expression given may not be in any of the above standard forms, especially when it involves functions that are raised to a power. In these cases, we use identities to transform them into standard forms. The identities often used are:

 

1. Fundamental identity

        sin²x + cos²x = 1 and the identities derived from there.

2. Double angle formulae, especially

       sin²x = 1- cos 2x

       cos²x = 1+ cos 2x

3. Factor formulae

 

Refer to our section on trigonometry for more details.

 

Solution:

 

(a)

              

 

(b)