- #1

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\int_{-\pi}^\pi \frac{\cos[(m+1)(x+\pi/2)]\sin[m(x+\pi/2)]\sin(x)}{\cos(x)}dx

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Here ##m## is positive integer. On this integral, I think ##\sin[m(x+\pi/2)]## is even function because it shifts by ##\pi/2##, the ##\cos(x)## and ##\cos[(m+1)(x+\pi/2)]## are also even functions except that ##\sin(x)## is an odd function, so the integrand is an odd function, the integral gives zero. I cannot find the result from the integral table and mathematica won't give me any number. So is my logic correct to get the zero?

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\int_{-\pi}^\pi \frac{\cos[(m+1)(x+\pi/2)]\sin[m(x+\pi/2)]}{\cos(x)}dx

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However, the similar reasoning fails in this case since all the even function. My note written that this integral should be zero but I cannot tell why.

I verify that those two integrals gives zero when m=1, 2, 3, ... 30 individually and manually. I didn't check the number after 30 since I am looking for a way to provide the general case for any m.