HSC Extension 1 and 2 Mathematics/3-Unit/HSC/Primitive of sin and cos squared

The integration of these functions can be performed by the substitution of or by a function in terms of respectively.

Deriving the substituted functionsEdit

By the double-angle formula,  . Rearranging to make   the subject,  .

We know by the Pythagorean identity   that  , so  . Rearranging,  .

Finding the primitiveEdit

By substituting   for  ,  . Similarly for  ,

 

These can be integrated by using the primitives of cosine and sine

 

 

For cos2 α

 

 

 

For sin2 α