TITLE: Prove that $\int_{0}^{\frac{\pi }{2}}\sin(\cos(x))+\cos(\sin(x))dx\leq \frac{\pi ^{2}}{4}$
QUESTION [12 upvotes]: Let $f:\left [ 0,\frac{\pi }{2} \right ]\rightarrow \mathbb{R}, f(x)=\sin(\cos(x))+\cos(\sin(x))$. Prove that $$\int\limits_{0}^{\pi/2}f(x)\,\mathrm{d}x\leq \frac{\pi ^{2}}{4}\,.$$

I've managed to find that $f$ is decreasing on $\left [ 0,\frac{\pi }{2} \right ]$, hence: 
$$
\int\limits_{0}^{\pi/2} f(x)\,\mathrm{d}x
     \leq \int\limits_{0}^{\pi/2} f(0)\,\mathrm{d}x
      =   \frac{\pi }{2}\cdot (1+\sin(1))$$ 
But $1+\sin(1)\geq \frac{\pi }{2}$, thus my inequality isn't enough to prove the problem statement.

REPLY [9 votes]: Notice $$\int_0^{\pi/2} \cos(\sin(x)) dx = \int_0^{\pi/2}\cos\left(\cos\left(\frac{\pi}{2}-x\right)\right) dx = \int_0^{\pi/2} \cos(\cos(x))dx$$
We have
$$\int_0^{\pi/2}\sin(\cos(x))+\cos(\sin(x))dx
= \int_0^{\pi/2}\sin(\cos(x)) + \cos(\cos(x)) dx\\
= \int_0^{\pi/2}\sqrt{2}\sin\left(\cos(x)+\frac{\pi}{4}\right) dx
\le \int_0^{\pi/2}\sqrt{2} dx =
\frac{\pi}{2}\sqrt{2}
\le \frac{\pi}{2}\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4}
$$