How to find factor (coefficient) in given function Fourier series?

Question

$$f(x)=x^3+\sin x$$ is given function. Interval is: $$(-π,π)$$ Fourier series is $$f(x)=\frac{a_{0}}{2}+\sum \limits_{n=1}^{\infty}(a_{n}\cos(nx)+b_{n}\sin(nx))$$ I have to find $$b_{5}$$

Answer 1

For all Fourier series of the form $$f(x)=\frac{a_{0}}{2}+\sum \limits_{n=1}^{\infty}(a_{n}\cos{(nx)}+b_{n}\sin{(nx)})$$ The values of $a_n,b_n$ are given by $$a_n=\frac1\pi\int_{-\pi}^\pi f(x)\cos{(nx)}dx$$ $$b_n=\frac1\pi\int_{-\pi}^\pi f(x)\sin{(nx)}dx$$ So for $b_5$ we need $$b_5=\frac1\pi\int_{-\pi}^\pi (x^3+\sin{(x)})\sin{(5x)}dx$$ Which can be seperated and integrated using integration by parts.