What is the integral of (e^{-5x}) ?

The integral of (e^{-5x}) is -1/5 e^{-5x}+C

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integral of (e^{-5x})

\int (e^{- 5 x}) d x

- \frac{1}{5} e^{- 5 x} + C

Solution steps

\int e^{- 5 x} d x

Apply u-substitution

= \int - \frac{1}{5} e^{u} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{5} e^{u}

Substitute back

u = - 5 x

= - \frac{1}{5} e^{- 5 x}

Add a constant to the solution

= - \frac{1}{5} e^{- 5 x} + C