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Step-by-step Solution

Integrate x^0.5(x^0.5+3)

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Answer

$\frac{1}{2}\left(\sqrt{x}+3\right)^{4}-4\left(\sqrt{x}+3\right)^{3}+9\left(\sqrt{x}+3\right)^2+C_0$

Step-by-step explanation

Problem to solve:

$\int\sqrt{x}\left(\sqrt{x}\:+3\:\right)dx$
1

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

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

Isolate $dx$ in the previous equation

$\frac{du}{\frac{1}{2}x^{-\frac{1}{2}}}=dx$

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Answer

$\frac{1}{2}\left(\sqrt{x}+3\right)^{4}-4\left(\sqrt{x}+3\right)^{3}+9\left(\sqrt{x}+3\right)^2+C_0$