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Example: the even case (alternate method II)

Evaluate:

Here we are integrating sine instead of cosine.

Everything works out as for cosine with sine and cosine interchanged and opposite signs for u and dv. As a result, the answer is the answer for cosine with sine and cosine interchanged and the opposite sign for all but the final x term:

Notice that this answer looks very different though it is the same as those obtained by the other methods.