What is the integral of e^{(-3x)} ?

The integral of e^{(-3x)} is -1/3 e^{-3x}+C

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integral of e^{(-3x)}

\int e^{(- 3 x)} d x

- \frac{1}{3} e^{- 3 x} + C

Solution steps

\int e^{- 3 x} d x

Apply u-substitution

= \int - \frac{1}{3} e^{u} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{3} e^{u}

Substitute back

u = - 3 x

= - \frac{1}{3} e^{- 3 x}

Add a constant to the solution

= - \frac{1}{3} e^{- 3 x} + C