What is the integral of x^3e^{-x} ?

The integral of x^3e^{-x} is -e^{-x}x^3-3(e^{-x}x^2-2(-e^{-x}x-e^{-x}))+C

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integral of x^3e^{-x}

\int x^{3} e^{- x} d x

- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})) + C

Solution steps

\int x^{3} e^{- x} d x

Apply u-substitution

= \int e^{u} u^{3} d u

Apply Integration By Parts

= u^{3} e^{u} - \int 3 u^{2} e^{u} d u

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - x

= (- x)^{3} e^{- x} - 3 ((- x)^{2} e^{- x} - 2 (e^{- x} (- x) - e^{- x}))

Simplify

(- x)^{3} e^{- x} - 3 ((- x)^{2} e^{- x} - 2 (e^{- x} (- x) - e^{- x})) : - e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))

= - e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))

Add a constant to the solution

= - e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x})) + C

What is the integral of x^3e^{-x} ?
The integral of x^3e^{-x} is -e^{-x}x^3-3(e^{-x}x^2-2(-e^{-x}x-e^{-x}))+C