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'=ex and so v=ex too. Then use the by parts formula to express the integral as x2ex-2(integral of):(xex-ex)dx. Using the by parts method again with the integral we just found we can reduce the second expression to x2ex-2xex+2ex. 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=x2ex between the points x=0 and x=5)
