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

The integral of e^{(-x)/a} is -ae^{-x/a}+C

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integral of e^{(-x)/a}

\int e^{\frac{- x}{a}} d x

- a e^{- \frac{x}{a}} + C

Solution steps

\int e^{\frac{- x}{a}} d x

Simplify

= \int e^{- \frac{x}{a}} d x

Apply u-substitution

: \int - a e^{u} d u

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

Take the constant out:

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

= - a \cdot \int e^{u} d u

Use the common integral:

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

= - a e^{u}

Substitute back

u = - \frac{x}{a}

= - a e^{- \frac{x}{a}}

Add a constant to the solution

= - a e^{- \frac{x}{a}} + C

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