Integrate e^(3x)^0.5

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

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Answer

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

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

Add the constant of integration

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

Answer

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

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Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.25 seconds

Views:

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