Saturday, 2 August 2014

calculus - Can one solve intinfty0fracsin(x)xdx *from its Taylor series antiderivative directly*?

This question was inspired by this question:




Evaluating the integral 0sinxx dx=π2?




Well, can anyone prove this without using Residue theory. I actually thought of doing this:
0sinxxdx=limtt01t(tt33!+t55!+)dt
but I don't see how \pi comes here, since we need the answer to be equal to \frac{\pi}{2}.




Answers were given to the stated question -- how to prove without using Residue theory. Yet the quote suggests an obvious follow-up question: can you prove the integral from the Taylor series expansion directly, somehow?

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