Trigonometric Integrals

Use the identities,

sin²x = 1/2 - 1/2 cos 2x

cos²x = 1/2 + 1/2 cos 2x

Proof of identities:

Starting with the double angle identity,      cos 2x = 2 cos²x - 1
                                                           cos 2x + 1 = 2 cos²x
                                                           cos 2x + 1 = cos ²x
                                                                 2
                                                  1/2 cos 2x + 1/2 = cos²x

Similarly, starting with the double angle identity,         cos 2x = 1 - 2 sin²x
                                                                             cos 2x - 1 = -2 sin²x
                                                                           cos 2x - 1 = sin²x
                                                                                   -2

1/2 - 1/2 cos 2x = sin²x

Use these identities to help simplify expressions to something for which an integral can be found.

Problems

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