A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:∞∫0sinxxdx=π2 Well, can anyone prove this without using Residue theory? I actually thought of using the series representation of sinx:∞∫0sinxxdx=limn→∞n∫01t(t−t33!+t55!+⋯)dtbut I don’t see how π comes ...Read more

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:∞∫0sinxxdx=π2

Well, can anyone prove this without using Residue theory? I actually thought of using the series representation of sinx:∞∫0sinxxdx=limn→∞n∫01t(t−t33!+t55!+⋯)dtbut I don’t see how π comes here, since we need the answer to be equal to π2.

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