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

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

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

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

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

Solution steps

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

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

: - e^{- s t} \frac{1}{s} (- 1 + t) - \int - e^{- s t} \frac{1}{s} d t

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

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

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

Add a constant to the solution

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

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