I am to calculate this: $$\int\limits_{0}^{1}dy\int\limits_{-\arccos(y)}^{\arccos(y)}e^{\sin(x)}dx$$ I noticed that $sin(x)$ is an odd function thus $e^{\sin(x)}$ is odd either but that's all. I have no further ideas how to deal with that integral.
2026-03-31 03:55:51.1774929351
A tricky multiple integral
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In this case, the trick is to switch the order of integration. If we rewrite the integral in the order $dxdy$, we have $$ \int_{-\pi/2}^{\pi/2} \int_0^{\cos(x)} e^{\sin(x)}\,dx\,dy $$ Switching the order of integration is a good thing to look for if you find that the first integral is not doable.