How do you integrate xln(x) between the limits of 0 and 2?

In order to answer this question you need to use integration by parts.Using the standard integration by parts formula: ∫u dv/dx dx = uv-∫v du/dx dx.Let:u=ln(x) v=(1/2)x²du/dx=1/x dv/dx=xTherefore we get:I=[1/2xln(x)-1/2∫xdx]²₀We now know how to integrate x. It becomes 1/2x². Therefore the overall integral becomes:I=[1/2xln(x)]²₀-[1/4x²]²₀I=2ln(2)-1I=ln(4/e)I ≈ 0.386