What is the integral of (-e^{-st})/s ?

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

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integral of (-e^{-st})/s

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

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

Solution steps

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

Take the constant out:

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

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

Simplify

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

Apply u-substitution

= (- \frac{1}{s}) \cdot \int - \frac{e ^{u}}{s} d u

Take the constant out:

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

= (- \frac{1}{s}) (- \frac{1}{s} \cdot \int e^{u} d u)

Use the common integral:

\int e^{u} d u = e^{u}

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

Substitute back

u = - s t

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

Simplify

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

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

Add a constant to the solution

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

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