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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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mengdie1982
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    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. – NewBornMATH Dec 11 '18 at 15:18
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    @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.! – mengdie1982 Dec 11 '18 at 15:24
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    Well ! You are welcome. You can upvote my comment if it helped. – NewBornMATH Dec 11 '18 at 15:25
  • Use https://math.stackexchange.com/questions/439851/evaluate-the-integral-int-frac-pi2-0-frac-sin3x-sin3x-cos3x/439856#439856 – lab bhattacharjee Dec 11 '18 at 16:01

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