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

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

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

Finding the primitive

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By substituting for , . Similarly for ,

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

For cos2 α

For sin2 α