How to prove the relations $\int_{0}^{+\infty}\sin(tu)\cos(t'u)du=pp\left(\displaystyle\frac{t}{t^2-t'^2}\right)$ and

Question

How to prove the relations:
$\int_{0}^{+\infty}\sin(tu)\sin(t'u)du=\displaystyle\frac{\pi}{2}[\delta(t-t')-\delta(t+t')]$
and
$\int_{0}^{+\infty}\sin(tu)\cos(t'u)du=pp\left(\displaystyle\frac{t}{t^2-t'^2}\right)$

Answer 1

I found the exact answer on the website called "the Fourier transform", in case anyone needs it as well I am posting the website as below: http://www.thefouriertransform.com/pairs/rightSidedSinusoids.php

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