What is the integral from 0 to infinity of x^8e^{-x} ?

The integral from 0 to infinity of x^8e^{-x} is 40320

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integral from 0 to infinity of x^8e^{-x}

\int_{0}^{\infty} x^{8} e^{- x} d x

40320

Solution steps

\int_{0}^{\infty} x^{8} e^{- x} d x

Compute the indefinite integral:

\int x^{8} e^{- x} d x = - e^{- x} x^{8} + 8 (- e^{- x} x^{7} - 7 (e^{- x} x^{6} - 6 (- e^{- x} x^{5} - 5 (e^{- x} x^{4} - 4 (- e^{- x} x^{3} - 3 (e^{- x} x^{2} - 2 (- e^{- x} x - e^{- x}))))))) + C

Compute the boundaries:

\int_{0}^{\infty} x^{8} e^{- x} d x = 0 - (- 40320)

= 0 - (- 40320)

Simplify

= 40320

What is the integral from 0 to infinity of x^8e^{-x} ?
The integral from 0 to infinity of x^8e^{-x} is 40320