MHB Evaluating $\int_{-1}^{1}\sin^7\left({x}\right) \,dx$ Using $u$-Substitution

[karush]

$\int_{-1}^{1}\sin^7\left({x}\right) \,dx$
$u=\cos\left({x}\right)$
$du=-\sin\left({x}\right)$
So
$-\int_{-1}^{1} \left(1-{u}^{2}\right)^3\,du$
So continue?

 

[MarkFL]

Your integrand is odd, and the limits symmetric about the origin, so can you think of a shortcut you can use?

 

[karush]

I presume you mean the areas will cancel out and the answer will be zero for interval given?

 

[MarkFL]

I presume you mean the areas will cancel out and the answer will be zero for interval given?

Yes, the odd-function rule can be directly applied here to state that the given definite integral has a value of 0. :D