TITLE: Compare the integrals $\int_0^{\frac{\pi}{2}}\sin(\cos x)dx$ and $\int_0^{\frac{\pi}{2}}\cos(\sin x)dx$
QUESTION [8 upvotes]: Compare the following two integrals:
$$\int_0^{\frac{\pi}{2}}\sin(\cos x)dx,\quad \int_0^{\frac{\pi}{2}}\cos(\sin x)dx$$
First I observe that by making the change of variable $x=\frac{\pi}{2}-x$,we have
$$\int_0^{\frac{\pi}{2}}\sin(\cos x)dx=\int_0^{\frac{\pi}{2}}\sin(\sin x)dx$$
Then I consider the function $f(x)=\sin(\sin x)-\cos(\sin x)$,after some simplification we have
$$f(x)=\frac{1}{\sqrt{2}}\sin(\sin x-\frac{\pi}{4})$$
Then I tried to determine the sign of $\int_0^{\frac{\pi}{2}}f(x)dx$ and I don't know how to proceed.

REPLY [10 votes]: since
use
$$\sin{x}\le x$$
so
$$\sin{(\cos{x})}\le\cos{x}$$
and
$y=\cos{x}$ is decreasing on $[0,\dfrac{\pi}{2}]$
so
$$\cos{x}\le\cos{(\sin{x})}$$
so
$$\sin{(\cos{x})}\le \cos{x}\le\cos{(\sin{x})}$$