$\int x^n \sin x dx$ I attempted solving this using parts and after using it twice I get:
$$(-x^n)\cos x + nx^{n-1}\sin x - n \int \sin x(n-1)x^{n-2} dx$$
can this indefinite integral be solved using typical under division Calculus I-II coursework?

Consider that:
$$ \int f(x)\,e^{-x}\,dx = C-\left(f(x)+f'(x)+f''(x)+\ldots\right)e^{-x}\tag{1}$$
and by setting $x=iy$:
$$\begin{eqnarray*}\int x^n\sin x\,dx &=& \text{Im}\int x^n e^{ix}\,dx = \text{Im}\left(i^{n+1}\int y^n e^{-y}\,dy\right)\tag{2}\end{eqnarray*}$$
from which it is easy to recover the final expression.
