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

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

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

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

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

Solution steps

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

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

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

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