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

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Answer 1 · 2018-04-18T19:04:14

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

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:

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

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