What is the integral of t^2*e^{-st} ?

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

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integral of t^2*e^{-st}

\int t^{2} \cdot e^{- s t} d t

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

Solution steps

\int 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

= - e^{- s t} \frac{1}{s} t^{2} - \int - e^{- s t} \frac{2}{s} t d t

\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}))

Simplify

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

Add a constant to the solution

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

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