How do you integrate #x^3e^x#?

1 Answer
George C.
Apr 18, 2018

#int \ x^3e^x \ dx = x^3e^x-3x^2e^x+6xe^x-6e^x+C#

Explanation:

Note that:

#d/(dx) (x^3e^x) = x^3e^x + 3x^2e^x#

#d/(dx) (-3x^2e^x) = -3x^2e^x-6xe^x#

#d/(dx) (6xe^x) = 6xe^x+6e^x#

#d/(dx) (-6e^x) = -6e^x#

So:

#d/(dx) (x^3e^x-3x^2e^x+6xe^x-6e^x) = x^3e^x#

So:

#int \ x^3e^x \ dx = x^3e^x-3x^2e^x+6xe^x-6e^x+C#

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