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

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

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integral of e^{-sx}x^2

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

x^{2} \cdot ln (7 x)

\int_{\frac{1}{5}}^{7} 10 x ln (5 x) d x

f (t) = sin (5 t)

\int 125 x^{2} cos (5 x) d x

\frac{\partial}{\partial u} (u^{2} v^{3})

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