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

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

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

\int_{0}^{π} t^{2} e^{- s t} d t

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

Solution steps

\int_{0}^{π} t^{2} e^{- s t} d t

Apply Integration By Parts

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

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

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

Simplify

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

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

Compute the boundaries:

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

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

Simplify

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

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