Evaluate $\int_0^\infty \frac{\sin s\cos s} s \,ds$ using $\int_0^\infty\frac{\sin s} s \, ds=\frac \pi 2$

Question

I got the answer $\frac{\pi}{4}$ by applying Fourier Integral Theorem but that didn't involve the given integral (although it did verify it). Any help would be appreciated.

Answer 1

Here's how it's supposed to work, I think: $$\int_0^\infty\frac{\sin s\cos s}{s}ds=\int_0^\infty\frac{\sin 2s}{2s}ds=\frac{1}{2}\int_0^\infty\frac{\sin s}{s}ds=\frac{\pi}{4}.$$