What is the integral of 2x^3e^{-x} ?

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

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integral of 2x^3e^{-x}

\int 2 x^{3} e^{- x} d x

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

Solution steps

\int 2 x^{3} e^{- x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{3} e^{- x} d x

Apply u-substitution

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

Apply Integration By Parts

= 2 (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}))

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

Substitute back

u = - x

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

Simplify

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

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

Add a constant to the solution

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

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