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

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

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

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

6

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

= 0 - (- 6)

Simplify

= 6

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