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

The integral from 0 to infinity of t*e^{-st} is 1/(s^2)

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

\int_{0}^{\infty} t \cdot e^{- s t} d t

\frac{1}{s ^{2}}

Solution steps

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

Compute the indefinite integral:

\int t e^{- s t} d t = \frac{1}{s ^{2}} (- s e^{- s t} t - e^{- s t}) + C

Compute the boundaries:

\int_{0}^{\infty} t e^{- s t} d t = 0 - (- \frac{1}{s ^{2}})

= 0 - (- \frac{1}{s ^{2}})

Simplify

= \frac{1}{s ^{2}}

What is the integral from 0 to infinity of t*e^{-st} ?
The integral from 0 to infinity of t*e^{-st} is 1/(s^2)