Evaluate the integral by parts

Question

no upper or lower bounds

9xsin(5x)dx

Answer 1

Your function has two pieces, 9x, and sin(5x). Since you know that you will be using integration by parts, what you have to decide is which piece you want to be u and which piece you want to be dv.

u gets differentiated in the integration by parts formula, so it's to your advantage to choose something that gets simpler when you differentiate it to be u. (9x)' = 9, which might seem a little simpler (it's just a number, rather than a linear function of x), but sin(5x)' = 5cos(5x), which is still a trig function. So it makes sense to choose u = 9x.

We have

u = 9x, du = 9 dx

dv = sin(5x) dx, v = -(1/5) cos(5x)

Then ∫ u dv = uv - ∫ v du = -(9/5) x cos(5x) - ∫ -(1/5) cos(5x) 9 dx

This simplifies a bit to -(9/5) x cos(5x) + (9/5) ∫ cos(5x) dx = -(9/5) x cos(5x) + (9/25) sin(5x) + c