What is the integral of x^2e^{-sx} ?

The integral of x^2e^{-sx} is -e^{-sx} 1/s x^2+2/(s^3)(-se^{-sx}x-e^{-sx})+C

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integral of x^2e^{-sx}

\int x^{2} e^{- s x} d x

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

Solution steps

\int x^{2} e^{- s x} d x

Apply Integration By Parts

: - e^{- s x} \frac{1}{s} x^{2} - \int - e^{- s x} \frac{2}{s} x d x

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

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

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

Simplify

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

Add a constant to the solution

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

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