Find the Fourier coefficients of $\cos^2(x)\sin^2(x)+2\cos^3(x)$

Question

So I know that the Fourier coefficients are expressed as:

$$a_0 = \frac{1}{\sqrt{2}\pi} \int_{-\pi}^{\pi}f(t)dt$$

$$c_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(kt)dt$$

$$b_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin(kt)dt$$

I know that I can plug these expressions into something like WolframAlpha and obtain solutions for each term, but I'm wondering if there's an easier way to do these without evaluating the integral. If I had such a question on an exam, it would be very tedious to work out the expressions by hand, so anything that makes things quicker would be great.

Answer 1

Note that $\sin^2(x)\cos^2(x) = \dfrac{\sin^2(2x)}4$ and $2\cos^3(x) = \dfrac{\cos(3x)+3\cos(x)}2$. Adding both gives you the appropriate Fourier series and therefore the Fourier coefficients.