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

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

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integral of e^{-st}

\int e^{- s t} d t

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

Solution steps

\int e^{- s t} d t

Apply u-substitution

: \int - \frac{e ^{u}}{s} d u

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

Use the common integral:

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

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

Substitute back

u = - s t

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

Add a constant to the solution

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

\frac{d}{d x} ((A x + B) e^{- x})

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f (x) = \frac{1}{8 x}

\frac{d}{d x} ((x + 1)^{- 1})

\int (2 x + 2 y) d y