Prove that $\int_{0}^{\frac{\pi }{2}}\sin(\cos(x))+\cos(\sin(x))dx\leq \frac{\pi ^{2}}{4}$
Solution 1:
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 $$\int_0^{\pi/2}\sin(\cos(x))+\cos(\sin(x))dx = \int_0^{\pi/2}\sin(\cos(x)) + \cos(\cos(x)) dx\\ = \int_0^{\pi/2}\sqrt{2}\sin\left(\cos(x)+\frac{\pi}{4}\right) dx \le \int_0^{\pi/2}\sqrt{2} dx = \frac{\pi}{2}\sqrt{2} \le \frac{\pi}{2}\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4} $$