Find the Fourier series of rectified sine wave $f(x) = \left( \sin x \right)^+$ (the non-negative part of sine function), i. e.,
$f(x) = \begin{cases} \sin x, \ \text{if} \sin x \geq 0.\\ 0, \ \text{if} \sin x < 0. \end{cases}$
$\textbf{My attempt:}$
$a_n = 0$ for $n \geq 0$, because $f$ is an odd function.
Let's calculate $b_n$.
$b_n = \frac{1}{\pi} \int^{\pi}_{-\pi} \sin (x) \sin(nx) dx$
If $u = \sin(nx)$ and $dv = \sin(x) dx$, then $du = n \cos (nx) dx$, $v = - cos (x)$ and
$\int \sin (x) \sin(nx) dx = - \sin (nx) \cos(x) - \int - \cos (x) n \cos (nx) dx$
$= - \sin (nx) \cos(x) + \int \cos (x) n \cos (nx) dx$
If $a = \cos (x)$ and $db = n \cos (nx) dx$, then $da = - \sin (x)$, $b = \sin (nx)$ and
$\int \cos (x) n \cos (nx) dx = \cos (x) \sin(nx) - \int \sin (nx) \left( - \sin (x) \right) dx$
$= \cos (x) \sin(nx) + \int \sin (nx) \sin (x) dx$
but now I can't compute $\int \sin (x) \sin (nx) dx$. I would like to know how to calculate this integral. Thanks in advance!