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HSC Mathematics Advanced, Extension 1, and Extension 2/3-Unit/HSC/Primitive of sin and cos squared
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The integration of these functions can be performed by the substitution of or by a function in terms of respectively.
Deriving the substituted functions
By the double-angle formula, . Rearranging to make the subject, .
We know by the Pythagorean identity that , so . Rearranging, .
Finding the primitive
By substituting for , . Similarly for ,
These can be integrated by using the primitives of cosine and sine
For cos2 α
For sin2 α