$\int_a^b x^n \sin(mx)dx$ Identity

Question

Are there nice memorable identity for the integrals

$$\int_a^b x^n \sin(mx)dx$$

$$\int_a^b x^n \cos(mx)dx$$

where n can be an integer from $0$ to $n$. When I try to derive something by integration by parts it gets awfully confusing, and I can't really find any nice patterns in the mess of my workings. I have a desire to be able to compute basic Fourier series quickly, at this time it takes me forever Thanks!

Answer 1

Consider at the same time the two integrals $I(n)$ and $J(n)$. Integrate, say, $I(n)$ by parts, using $u = x^n$, and on the remaining integral you will see $J(n-1)$ appearing. If I am correct, you will get

$$I(n) = \frac{- x^n\cos[m x] + n \cdot J(n-1)}m .$$

In the same way, integrate $J(n)$ by parts, using $u = x^n$, and on the remaining integral you will see $I(n-1)$ appearing. If I am correct, you will get

$$J(n) = \frac{x^n\sin[m x] - n \cdot I(n-1)}m .$$

These are very simple recurrence relations. Is this what you are looking for ?