What is the integral of (1-t)e^{-st} ?

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

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integral of (1-t)e^{-st}

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

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

Solution steps

\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

= - 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}}

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

Simplify

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

Add a constant to the solution

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

x \to - 2 lim (x^{2} - 2)

\frac{d}{d x} (x^{\frac{1}{5}} + 4)

\frac{d y}{d x} + y = \frac{1}{e ^{2 x} - 5 e ^{x} + 4}

(3 x - 2)^{\frac{3}{2}}

\frac{d}{d x} (e^{3 x} + 3 e^{3 x} x)

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