What is the integral from 0 to infinity of x^3e^{-x} ?

The integral from 0 to infinity of x^3e^{-x} is 6

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integral from 0 to infinity of x^3e^{-x}

\int_{0}^{\infty} x^{3} e^{- x} d x

6

Solution steps

\int_{0}^{\infty} x^{3} e^{- x} d x

Compute the indefinite integral:

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

Compute the boundaries:

\int_{0}^{\infty} x^{3} e^{- x} d x = 0 - (- 6)

= 0 - (- 6)

Simplify

= 6

What is the integral from 0 to infinity of x^3e^{-x} ?
The integral from 0 to infinity of x^3e^{-x} is 6