What is the integral of (20)/(e^{x/5)} ?

The integral of (20)/(e^{x/5)} is -100e^{-x/5}+C

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integral of (20)/(e^{x/5)}

\int \frac{20}{e ^{\frac{x}{5}}} d x

- 100 e^{- \frac{x}{5}} + C

Solution steps

\int \frac{20}{e ^{\frac{x}{5}}} d x

Take the constant out:

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

= 20 \cdot \int \frac{1}{e ^{\frac{x}{5}}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{e ^{\frac{x}{5}}} = e^{- \frac{x}{5}}

= 20 \cdot \int e^{- \frac{x}{5}} d x

Apply u-substitution

: \int - 5 e^{u} d u

= 20 \cdot \int - 5 e^{u} d u

Take the constant out:

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

= 20 (- 5 \cdot \int e^{u} d u)

Use the common integral:

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

= 20 (- 5 e^{u})

Substitute back

u = - \frac{x}{5}

= 20 (- 5 e^{- \frac{x}{5}})

Simplify

= - 100 e^{- \frac{x}{5}}

Add a constant to the solution

= - 100 e^{- \frac{x}{5}} + C

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What is the integral of (20)/(e^{x/5)} ?
The integral of (20)/(e^{x/5)} is -100e^{-x/5}+C