Integrate (x/2+3)^2

\int\left(\frac{x}{2}+3\right)^2dx

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Answer

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

Step by step solution

Problem

$\int\left(\frac{x}{2}+3\right)^2dx$
1

Solve the integral $\int\left(3+\frac{x}{2}\right)^2dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=3+\frac{x}{2} \\ du=\frac{1}{2}dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{\frac{1}{2}}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{u^2}{\frac{1}{2}}du$
4

Taking the constant out of the integral

$2\int u^2du$
5

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

$2\frac{u^{3}}{3}$
6

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

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

Simplify the fraction

$\frac{2}{3}\left(3+\frac{x}{2}\right)^{3}$
8

Add the constant of integration

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

Answer

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

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.5 seconds

Views:

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