How do you find the integral of 'x sin(2x) dx'?

To begin, notice that the expression involves 2 functions of x multiplied together, this means we can approach the problem using integration by parts. Integration by parts can be expressed as '∫ u (dv/dx) dx = uv - ∫ v(du/dx) dx'. In order to apply this to our problem, we must first break up our expression into each function. u = x dv/dx = sin(2x) dx (LHS of 'by parts' equation) For the next step, we must; 1. differentiate u to find du/dx 2. integrate dv/dx u = x → du/dx = 1 dv/dx → v = -0.5cos(2x) we now have expressions for u, v, du/dx and dv/dx. Plug in these expressions to the RHS of the 'by parts' equation giving = (x)(-0.5cos(2x)) - ∫(1)(-0.5)cos(2x) integrate (x)(-0.5cos(2x)) - (-0.25)sin(2x) simplify ∫x sin(2x) dx = 0.25sin(2x) - 0.5xcos(2x) *What to watch for. When deciding which function of x to assign to u or dv/dx, try to pick the function that will differentiate to the simplest function for u. In some cases, e.g. if ln(x) is involved, this can't be directly integrated and therefore should assigned to u.