Numerical integration, number sort,$\int \sin \left(\sin \left(x\right)\right)dx$

Question

The blue line is the value of the lookup (the method is similar to the punctuation in Figure 2). Is there a smarter way?

Answer 1

For $x\in[0,\,\pi/4]$, $0\le\sin x\le x\le\tan x\le 1$, so $0\le\sin(\sin x)\le\sin x\le\sin(\tan x)\le\sin 1$, whence $J<I<K$.