Integral of (3x+2)^0.5

\int\sqrt{3x+2}dx

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Answer

$\frac{2}{9}\sqrt{\left(2+3x\right)^{3}}+C_0$

Step by step solution

Problem

$\int\sqrt{3x+2}dx$
1

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

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

Isolate $dx$ in the previous equation

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

Substituting $u$ and $dx$ in the integral

$\int\frac{\sqrt{u}}{3}du$
4

Taking the constant out of the integral

$\frac{1}{3}\int\sqrt{u}du$
5

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{1}{3}\cdot \frac{2}{3}\sqrt{u^{3}}$
6

Substitute $u$ back for it's value, $2+3x$

$\frac{2}{9}\sqrt{\left(2+3x\right)^{3}}$
7

Add the constant of integration

$\frac{2}{9}\sqrt{\left(2+3x\right)^{3}}+C_0$

Answer

$\frac{2}{9}\sqrt{\left(2+3x\right)^{3}}+C_0$

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Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.25 seconds

Views:

186