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Probem

Evaluate $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^3 x\cdot f(\cos x){\rm d}x.$$

Thanks to @NewBornMATH's hint. It's easy to verify that the integrand is an odd function,and the integral interval is symmetric with respect to the original point. Hence the integral value must be zero!

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closed as off-topic by MisterRiemann, Saad, Nosrati, Dando18, Brahadeesh Dec 12 '18 at 8:26

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    $\begingroup$ From the even odd property it seems to be 0. Consider $g(x)=(sin(x))^{3}.f(cosx)$ then note that $g(-x)=-g(x)$ hence integral from any $(-a,a)$ will be zero. $\endgroup$ – NewBornMATH Dec 11 '18 at 15:18
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    $\begingroup$ @NewBornMATH Oh,yes!It's easy to verify that the integrand is an odd function,and the integral interval is symmetric with the original point. Hence the Integral value must be zero! thanks.! $\endgroup$ – mengdie1982 Dec 11 '18 at 15:24
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    $\begingroup$ Well ! You are welcome. You can upvote my comment if it helped. $\endgroup$ – NewBornMATH Dec 11 '18 at 15:25
  • $\begingroup$ Use math.stackexchange.com/questions/439851/… $\endgroup$ – lab bhattacharjee Dec 11 '18 at 16:01