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

The integral of e^{-kx} is -1/k e^{-kx}+C

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integral of e^{-kx}

\int e^{- k x} d x

- \frac{1}{k} e^{- k x} + C

Solution steps

\int e^{- k x} d x

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = - k x

= - \frac{1}{k} e^{- k x}

Add a constant to the solution

= - \frac{1}{k} e^{- k x} + C