real analysis - Is there a general formula for $int_0^l x^n sin(mpi x/l) dx$?

The integral

$$\int_0^l x^n \sin\left(\frac{m\pi x}l\right) dx$$
frequently arises for computing Fourier coefficients, for $m,n$ integers. Is there any general formula for that? What about $\cos$ instead of $\sin$?

Answer

$$J_{\sin}=\int_0^lx^n\sin\left(\dfrac{m\pi x}{l}\right)dx$$
$$J_{\sin}=\dfrac{1}{n+2}\left(\pi ml^{n+1}F^1_2\left(\dfrac{n}{2}+1;\dfrac{3}{2},\dfrac{n}{2}+2;-\dfrac{1}{4}m^2\pi^2\right)\right)$$
with $$\Re(n)\gt 2$$
and $$F^p_q(a_1...a_p;b_1...b_q;z)$$
is the generalized Hypergeometric function.

$$J_{\cos}=\int_0^lx^n\cos\left(\dfrac{m\pi x}{l}\right)dx$$
$$J_{\cos}=\dfrac{1}{n+2}\left(l^{n+1}F^1_2\left(\dfrac{n}{2}+\dfrac{1}{2};\dfrac{1}{2},\dfrac{n}{2}+\dfrac{3}{2};-\dfrac{1}{4}m^2\pi^2\right)\right)$$
$$\Re(n)\gt -1$$