How to $\int_{-1}^{1} \int_{-1}^{1} \sin^{2019}\,(xy)\, dx\, dy$

Question

$$ \int_{-1}^{1} \int_{-1}^{1} \sin^{2019}(xy) \,dx \,dy$$

I need to calculate this double integral but since the sine function is to the so high power, I'm not sure how to calculate this one. If you could help me and maybe explain how to deal, if there is a way, with high powers of trigonometric functions under integral.

You could start with https://math.stackexchange.com/questions/112687/integrating-int-sinnx-dx — Matti P., Nov 26 '19 at 11:42
Or note that the integrand is odd and the region is symmetric about the origin. — lulu, Nov 26 '19 at 11:44
Try with the definition of $\sin u$ in terms of complex exponential. — Bernard, Nov 26 '19 at 11:46

score 1 · Accepted Answer · answered Nov 26 '19 at 11:50

1

I am just trying to explain what Lulu said. Note that for any fixed $y$ the function $sin^{2019}(xy)$ is an odd function, and hence it's integral is $0$ on $[-1, 1]$. So, we're done.

answered Nov 26 '19 at 11:50

Matha Mota

3,472

How to $\int_{-1}^{1} \int_{-1}^{1} \sin^{2019}\,(xy)\, dx\, dy$

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