Definite Integral of $\sin(\sin x)$

Question

I recently came across this problem:

Prove that $\int_{0}^{\frac{\pi}{2}} \sin(\sin y) dy \leq 1$.

However, from my understanding, the expression $\sin(\sin y)$ does not have an indefinite integral (How to find the indefinite integral of sin(sin(x))dx?). So, how will we go about solving the above problem using just the Riemann sums?

Answer 1

Define $f(x) = \sin(\sin x)$. We know that $0 \le \sin x \le x$. As $\sin x$ is increasing on $[0, \pi/2]$, $0 \le f(x) \le \sin x$ on this interval.

Taking the integral, you're done.