Prove
∫∞0|sinx|sinxxdx=1.
I know how to calculate ∫∞0sinxxdx=π2, but the method cannot be applied here. So I am thinking
n∑k=0(−1)k∫(k+1)πkπsin2xxdx
but I don't know how to proceed.
Answer
By Lobachevsky integral formula: https://en.wikipedia.org/wiki/Lobachevsky_integral_formula
∫∞0sinxx|sinx|dx=∫π/20|sinx|dx=1.
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