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)

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