What is the integral from 0 to 1 of t^2e^{-st} ?

The integral from 0 to 1 of t^2e^{-st} is -(e^{-s})/s+(2(-se^{-s}-e^{-s})+2)/(s^3)

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integral from 0 to 1 of t^2e^{-st}

\int_{0}^{1} t^{2} e^{- s t} d t

- \frac{e ^{- s}}{s} + \frac{2 ( - s e ^{- s} - e ^{- s} ) + 2}{s ^{3}}

Solution steps

\int_{0}^{1} t^{2} e^{- s t} d t

Apply Integration By Parts

: [- e^{- s t} \frac{1}{s} t^{2} - \int - e^{- s t} \frac{2}{s} t d t]_{0}^{1}

= [- e^{- s t} \frac{1}{s} t^{2} - \int - e^{- s t} \frac{2}{s} t d t]_{0}^{1}

\int - e^{- s t} \frac{2}{s} t d t = - \frac{2}{s ^{3}} (- s e^{- s t} t - e^{- s t})

= [- e^{- s t} \frac{1}{s} t^{2} - (- \frac{2}{s ^{3}} (- s e^{- s t} t - e^{- s t}))]_{0}^{1}

Simplify

= [- e^{- s t} \frac{1}{s} t^{2} + \frac{2}{s ^{3}} (- s e^{- s t} t - e^{- s t})]_{0}^{1}

Compute the boundaries:

- e^{- s} \frac{1}{s} + \frac{2}{s ^{3}} (- s e^{- s} - e^{- s}) + \frac{2}{s ^{3}}

= - e^{- s} \frac{1}{s} + \frac{2}{s ^{3}} (- s e^{- s} - e^{- s}) + \frac{2}{s ^{3}}

Simplify

= - \frac{e ^{- s}}{s} + \frac{2 ( - s e ^{- s} - e ^{- s} ) + 2}{s ^{3}}

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What is the integral from 0 to 1 of t^2e^{-st} ?
The integral from 0 to 1 of t^2e^{-st} is -(e^{-s})/s+(2(-se^{-s}-e^{-s})+2)/(s^3)