What is the integral of 3e^{-9x} ?

The integral of 3e^{-9x} is -1/3 e^{-9x}+C

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integral of 3e^{-9x}

\int 3 e^{- 9 x} d x

- \frac{1}{3} e^{- 9 x} + C

Solution steps

\int 3 e^{- 9 x} d x

Take the constant out:

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

= 3 \cdot \int e^{- 9 x} d x

Apply u-substitution

: \int - \frac{1}{9} e^{u} d u

= 3 \cdot \int - \frac{1}{9} e^{u} d u

Take the constant out:

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

= 3 (- \frac{1}{9} \cdot \int e^{u} d u)

Use the common integral:

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

= 3 (- \frac{1}{9} e^{u})

Substitute back

u = - 9 x

= 3 (- \frac{1}{9} e^{- 9 x})

Simplify

3 (- \frac{1}{9} e^{- 9 x}) : - \frac{1}{3} e^{- 9 x}

= - \frac{1}{3} e^{- 9 x}

Add a constant to the solution

= - \frac{1}{3} e^{- 9 x} + C

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