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

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

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integral of 1/θ e^{-x/θ}

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

- e^{- \frac{x}{θ}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - θ e^{u} d u

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{θ} (- θ e^{u})

Substitute back

u = - \frac{x}{θ}

= \frac{1}{θ} (- θ e^{- \frac{x}{θ}})

Simplify

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

= - e^{- \frac{x}{θ}}

Add a constant to the solution

= - e^{- \frac{x}{θ}} + C

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