I have to determine the value of $\int_{-\pi }^{\pi } \left(\cos (3\cdot x)\cdot \sin (x)^2\right)^2 \, dx$.
What I did was rewriting the integrand as $$\frac{1}{16}(6\cdot \sin (x)-\sin (3\cdot x)-3\cdot \sin (5\cdot x)+3\cdot \sin (7\cdot x)-\sin (9\cdot x) )$$
And then using the fourier coefficient to find $3/8\cdot\pi$.
But I believe this is no way near the right method, because it includes the expansion. How to do this simpler?