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

The integral of (e^{3x})/(e^x) is 1/2 e^{2x}+C

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

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

\frac{1}{2} e^{2 x} + C

Solution steps

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

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{3}{2} u^{\frac{2}{3}}

= \frac{1}{3} \cdot \frac{3}{2} u^{\frac{2}{3}}

Substitute back

u = e^{3 x}

= \frac{1}{3} \cdot \frac{3}{2} (e^{3 x})^{\frac{2}{3}}

Simplify

\frac{1}{3} \cdot \frac{3}{2} (e^{3 x})^{\frac{2}{3}} : \frac{1}{2} e^{2 x}

= \frac{1}{2} e^{2 x}

Add a constant to the solution

= \frac{1}{2} e^{2 x} + C

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What is the integral of (e^{3x})/(e^x) ?
The integral of (e^{3x})/(e^x) is 1/2 e^{2x}+C