I have a function F, and I want to test whether it is an increasing function.

$F(x)=\int_0^x e^{-\sin(t)}dt$

Problem: I tried to solve the above integral(using parts) and it seems it is a series, since it's expopnential. How can I solve the above integral?

The integration by parts goes on for another two times, and I still ended up with a series. Any easy way of checking whether it is an incresing function?.

My Approach: For normal functions(that are not integral), I would use $F'(x)=0$ and check the increasing nature by equating with the limits.

$F(x)=\int_0^x e^{-\sin(t)}dt F(x)=\int_0^x e^{-u}\frac{du}{\sqrt{1-u^{2}}}$

$u=sin(t);\ du=cos(t)dt \implies dt\frac{du}{\sqrt{cos(t)}} \implies dt=\frac{du}{\sqrt{1-u^{2}}}$


$f(x) = e^{-\sin x}$ is always positive, so the area under its graph must be always increasing.

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