Fourier serier with complex integral

Question

I have a problem with a long calculus of Fourier series coefficient.

The integral is $\int_{-\pi}^{\pi} x^4 cos{nx} dx$.

How can resolve quickly with $\int_{-\pi}^{\pi} x^4 e^{-inx} dx$ instead of integral by parts??

Thank you

Answer 1

HINT:

$$\int_a^b x^4\cos(nx)dx=\text{Re}\int_a^b x^4e^{-inx}dx=\text{Re}\frac{\partial^4}{\partial n^4}\int_a^b e^{-inx}dx$$ and $$\int^\pi_{-\pi}e^{-inx}dx=\frac{e^{-\pi ni}-e^{\pi ni}}{-ni}=\frac2n \sin(n\pi)$$