Finding the RMS value

Question

Here, I can't understand the third step. Why they ignored other three terms \$(40000\sin\omega t\sin 3\omega t, 10000\sin 3\omega t\sin 5\omega t, 20000\sin 5\omega t\sin\omega t)?\$ Please help.

Answer 1

Replace \$\sin(A)\sin(B)\$ with \$\frac{1}{2}(\cos(A-B)-\cos(A+B))\$. (Reference) Perform the integration after doing the above substitution. It will be zero since area under the curve of \$\cos(\cdot)\$ is zero over the range of integration performed.