What is the integral of e^{-αx} ?

The integral of e^{-αx} is -1/α e^{-αx}+C

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integral of e^{-αx}

\int e^{- αx} d x

- \frac{1}{α} e^{- αx} + C

Solution steps

\int e^{- αx} d x

Apply u-substitution

= \int - \frac{e ^{u}}{α} d u

Take the constant out:

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

= - \frac{1}{α} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{α} e^{u}

Substitute back

u = - αx

= - \frac{1}{α} e^{- αx}

Add a constant to the solution

= - \frac{1}{α} e^{- αx} + C