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

The integral of-3e^{-x/3} is 9e^{-x/3}+C

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integral of-3e^{-x/3}

\int - 3 e^{- \frac{x}{3}} d x

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

Solution steps

\int - 3 e^{- \frac{x}{3}} d x

Take the constant out:

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

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

Apply u-substitution

: \int - 3 e^{u} d u

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

Take the constant out:

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

= - 3 (- 3 \cdot \int e^{u} d u)

Use the common integral:

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

= - 3 (- 3 e^{u})

Substitute back

u = - \frac{x}{3}

= - 3 (- 3 e^{- \frac{x}{3}})

Simplify

= 9 e^{- \frac{x}{3}}

Add a constant to the solution

= 9 e^{- \frac{x}{3}} + C

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