What is the integral of 1/a e^{-x/a} ?

The integral of 1/a e^{-x/a} is -e^{-x/a}+C

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integral of 1/a e^{-x/a}

\int \frac{1}{a} e^{- \frac{x}{a}} d x

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

Solution steps

\int \frac{1}{a} e^{- \frac{x}{a}} d x

Take the constant out:

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

= \frac{1}{a} \cdot \int e^{- \frac{x}{a}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = - \frac{x}{a}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 1/a e^{-x/a} ?
The integral of 1/a e^{-x/a} is -e^{-x/a}+C