What is the integral of (e^{2x})/(e^{x+3)} ?

The integral of (e^{2x})/(e^{x+3)} is e^{x-3}+C

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integral of (e^{2x})/(e^{x+3)}

\int \frac{e ^{2 x}}{e ^{x + 3}} d x

e^{x - 3} + C

Solution steps

\int \frac{e ^{2 x}}{e ^{x + 3}} d x

Simplify

\frac{e ^{2 x}}{e ^{x + 3}} : e^{x - 3}

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

e^{x - 3} = e^{- 3} e^{x}

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

Take the constant out:

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

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

Apply exponent rule:

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

= \frac{1}{e ^{3}} \cdot \int e^{x} d x

Use the common integral:

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

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

Simplify

\frac{1}{e ^{3}} e^{x} : e^{x - 3}

= e^{x - 3}

Add a constant to the solution

= e^{x - 3} + C

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What is the integral of (e^{2x})/(e^{x+3)} ?
The integral of (e^{2x})/(e^{x+3)} is e^{x-3}+C