Find the integral of xcos(2x) with respect to x

You can see that this question is asking you to do integration by parts. Remember that the integral of uv' is equal to uv - the integral of u'v. You want to find a u that gets easier when you differentiate it and a v' that's possible to integrate directly and doesn't get messier when you integrate it. In this case let u = x and v' = cos(2x). u' = 1 and v = sin(2x)/2. The integral of xcos(2x) = xsin(2x)/2 - the integral of sin(2x)/2Hence the integral of xcos(2x) = xsin(2x)/2 + cos(2x)/4 + c.