Integrate x^2e^x with respect to x between the limits of x=5 and x=0.

This question is to test integration by parts.  First let u=x² and u'=2x as a result, and v'=e^x and so v=e^x too. Then use the by parts formula to express the integral as x²e^x​​​​​​​-2(integral of):(xe^x​​​​​​​-e^x​​​​​​​)dx. Using the by parts method again with the integral we just found we can reduce the second expression to x²e^x​​​​​​​-2xe^x​​​​​​​+2e^x​​​​​​​. This is the indefinite integration result of the expression in question, (without the +c!) and the question is finished by first plugging 5 into the answer, to get 25e⁵-10e⁵​+2e⁵​ and then plugging in 0, to give just 2. Subtracting the lower limit from the upper limit results gives 25e⁵​-10e⁵​+2e⁵​-2, which is a constant term and is the answer to the question (in physical terms it is the area underneath the curve y=x²e^x between the points x=0 and x=5)