Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ [duplicate]

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ [duplicate]

This question already has an answer here:
What is the odd fourier extention of sin x cos(2x) 1 answer
Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ in the range
$ -\pi \leq x \leq \pi $
any help much appreciated
I need find out
$a_0$ and $a_1$ and $b_1$
I can find $a_0$ which is simply integrating something with respect to the
limits I can get as far as
$$\frac{1}{2} \int_{-\pi}^\pi \ \frac12 (\sin (3x)-\sin(x)) dx$$
How would I integrate the above expression ?
secondly how would I calculate $a_1$ and $b_1$ but despite knowing the
general formula to find the fourier series Im having trouble applying them
to this question