Question

Evaluate using Integration by Parts.
x^2 cos(3x) dx

Answer 1

Integration by parts is just the product rule in reverse:
d(uv) = u dv + v du

∫u dv = uv - ∫ v du
Let
u = x^2
du = 2x dx

dv = cos(3x) dx
v = 1/3 sin(3x)

∫x^2 cos(3x) dx
= (x^2)(1/3 sin(3x)) - (2/3)∫x sin(3x)
dx

Let
u = x
du = dx

dv = sin(3x) dx
v = -1/3 cos(3x)

∫x sin(3x) dx
= (x)(-1/3 cos(3x)) + ∫-1/3 cos(3x) dx

Now just put it all together. You can verify your answer at wolframalpha.com