Question

evaluate the integral.
from 0 to 4π
t^2 sin(2t) dt

Answer 1

you will need to use integration by parts, twice.

1st step:
u = t^2
dv=sin(2t) dt

du = 2t dt
v = -1/2 cos(2t)

That makes the first integration by parts: ?u dv = uv - ?v du

?t^2 sin(2t) dt
= -1/2 t^2 cos(2t) + ?t cos(2t) dt

Then repeat, letting
u = t
dv = cos(2t) dt

You should wind up with this:

http://www.wolframalpha.com/input/?i=%E2%88%AB+t%5E2+sin(2t)+dt

Then evaluate that at the limits.