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

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

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integral of e^{-st}(t-2)

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

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

Solution steps

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

Expand

(- 2 + t) e^{- s t} : - 2 e^{- s t} + e^{- s t} t

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

Add a constant to the solution

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

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