Integrate xsin(x).

The technique we need to use to solve this integral is called integration by parts. The parts formula is: the integral of (uv' dx) = uv - the integral of (u'v dx) (where u and v are functions of x). We need to decide which of our functions (x or sin(x)) is our u and which is our v'. To pick our 'u' we consider which function becomes simpler when we differentiate it. In this case this is x since its derivative is 1 whereas the derivative of sin(x) is cos(x) which isn't much simpler. So u = x, v' = sin(x). Which means u' = 1 , v = -cos(x). So our integral becomes: -xcos(x) - the integral of (-cos(x)dx). Giving our final answer of : -xcos(x) + sin(x) + c