What is the integral from 1 to infinity of e^{-st}t ?

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

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integral from 1 to infinity of e^{-st}t

\int_{1}^{\infty} e^{- s t} t d t

- \frac{- e ^{- s} s - e ^{- s}}{s ^{2}}

Solution steps

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

= 0 - \frac{- e ^{- s} s - e ^{- s}}{s ^{2}}

Simplify

= - \frac{- e ^{- s} s - e ^{- s}}{s ^{2}}

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