Integrate xcos(x) with respect to x

In order to do this question one has to first recognise that this is a product and thus has to be solved using Integration by parts. Firstly write out the formula: ∫u.dv/dx dx = uv - ∫v. du/dx dx (this is always provided in the formula booklet) Now we have the formula we have to assign values to each of the variables and their derivatives. Remeber that unless we are dealing with ln(x) we always assign u=x and therefore in this scenario dv/dx = cos(x). Next we calulate v and du/dx. As dv/dx = cos(x) we know v = sin(x) (as sin(x) is the integral of cos(x)), and the du/dx is simply 1. Now we simply sub these into our formula so therfore ∫xcosx dx = xsin(x) - ∫sin(x) dx We know the integral of sin(x) is simply -cos(x) and therefore our answer is xsin(x) + cos(x) + C (always remembering the constant as this will differentiate to 0) TIPS: I always like to write in a separate box the values I assign to the variables so as to keep it away from the main argument and to keep it tidy. To check you've done the integration correctly simply differentiate using the product rule. You should get the original equation.