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

The integral of e^{-bt} is -1/b e^{-bt}+C

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integral of e^{-bt}

\int e^{- b t} d t

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

Solution steps

\int e^{- b t} d t

Apply u-substitution

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

= \int - \frac{e ^{u}}{b} d u

Take the constant out:

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

= - \frac{1}{b} \cdot \int e^{u} d u

Use the common integral:

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

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

Substitute back

u = - b t

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

Add a constant to the solution

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

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