Evaluate the integral between 5 and 3 for xe^x

Firstly, by looking at the integral, we see that this will have to be done by parts. There are 2 parts of this expression, the 'x' and the 'e^x'. As we know the formula for integration by parts is: integral of u dv/dx = uv - integral of v du/dx, we must assign u and dv/dx to each of the 2 parts of our expression. The final result we want to find is the integral of u dv/dx, as this is what the question wants us to find. As e^x integrates and differentiates to itself, it is easier to assign this to dv/dx as this is the part of the expression we integrate. Therefore, we must assign x to u. To obtain the remaining 2 parts, du/dx and v, we must integrate dv/dx and differentiate u. This gives us du/dx=1 and v=e^x.Now we have all the components to sub into our formula, we get that xe^x - integral of e^x. So the final expression is e^x(x-1). From here, it is straightforward to evaluate this between 5 and 3, and it gives us:2x^3(2x^2 - 1).As for integrals like these, examination papers will normally ask you to give an exact answer, do not evaluate this further as it will lead to a very Long decimal and loss of marks