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

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

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integral of sqrt(e^{3x)}

\int e^{3 x} d x

\frac{2}{3} e^{\frac{3 x}{2}} + C

Solution steps

\int e^{3 x} d x

Apply radical rule:

assuming

a \geq 0

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

Apply u-substitution

: \int e^{u} \frac{2}{3} d u

= \int e^{u} \frac{2}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{2}{3} e^{u}

Substitute back

u = \frac{3 x}{2}

= \frac{2}{3} e^{\frac{3 x}{2}}

Add a constant to the solution

= \frac{2}{3} e^{\frac{3 x}{2}} + C

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