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

The integral of-ke^{-kx} is e^{-kx}+C

integral of-ke^{-kx}

\int - k e^{- k x} d x

e^{- k x} + C

Solution steps

\int - k e^{- k x} d x

Take the constant out:

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

= - k \cdot \int e^{- k x} d x

Apply u-substitution

= - k \cdot \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

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

Use the common integral:

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

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

Substitute back

u = - k x

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

Simplify

- k (- \frac{1}{k} e^{- k x}) : e^{- k x}

= e^{- k x}

Add a constant to the solution

= e^{- k x} + C

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