# Integrate e^(3x)^0.5

## \int\sqrt{e^{3x}}dx

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e
π
ln
log
lim
d/dx
d/dx
>
<
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<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

$\frac{2}{3}e^{\frac{3}{2}x}+C_0$

## Step by step solution

Problem

$\int\sqrt{e^{3x}}dx$
1

Applying the power of a power property

$\int e^{\frac{3}{2}x}dx$
2

Solve the integral $\int e^{\frac{3}{2}x}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=\frac{3}{2}x \\ du=\frac{3}{2}dx\end{matrix}$
3

Isolate $dx$ in the previous equation

$\frac{du}{\frac{3}{2}}=dx$
4

Substituting $u$ and $dx$ in the integral

$\int\frac{e^u}{\frac{3}{2}}du$
5

Taking the constant out of the integral

$\frac{2}{3}\int e^udu$
6

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$\frac{2}{3}e^u$
7

Substitute $u$ back for it's value, $\frac{3}{2}x$

$\frac{2}{3}e^{\frac{3}{2}x}$
8

$\frac{2}{3}e^{\frac{3}{2}x}+C_0$

$\frac{2}{3}e^{\frac{3}{2}x}+C_0$

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### Main topic:

Integration by substitution

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