# 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!

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – MisterRiemann, Saad, Nosrati, Dando18, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.

• 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
• @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
• Well ! You are welcome. You can upvote my comment if it helped. – NewBornMATH Dec 11 '18 at 15:25
• – lab bhattacharjee Dec 11 '18 at 16:01