$\int x^n \sin x dx$ 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?

Define $$ I_n = \int x^n\sin(x)dx $$
$$U_n = \int x^n\cos(x)dx$$
Integration by parts 
$$ I_n =-x^n\cos(x)+n\int x^{n-1}\cos(x)dx=-x^n\cos(x)+nU_{n-1}$$
$$U_n=x^n\sin(x)-n\int x^{n-1}\sin(x)dx=x^n\sin(x)-nI_{n-1}$$
with 
$$I_0= -\cos(x)$$
$$U_0=\sin(x)$$
Now substitute until you get to $I_0$ or $U_0$ depending on the parity of $n$.
