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

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

integral of e^{-st}t^2

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

Apply Integration By Parts

= - 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

\int \frac{x}{e ^{2 x}} d x

x \to (- 1 + 1) lim (\frac{1}{x - 2})

\int x cos (x) + cos (x) - 1 d x

\int cos^{2} (x) - sin^{2} (x) d x

l a pl a ce t r an s f or m (t - 3)^{4}

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