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Step-by-step Solution

Integral of x(x+3)^(1/2)

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Answer

$\frac{2}{5}\sqrt{\left(x+3\right)^{5}}-2\sqrt{\left(x+3\right)^{3}}+C_0$

Step-by-step explanation

Problem to solve:

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

Solve the integral $\int x\sqrt{x+3}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=3\tan\left(\theta\right)^{2} \\ dx=6\sec\left(\theta\right)^2\tan\left(\theta\right)d\theta\end{matrix}$
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Substituting in the original integral, we get

$\int18\tan\left(\theta\right)^{2}\sqrt{3\tan\left(\theta\right)^{2}+3}\sec\left(\theta\right)^2\tan\left(\theta\right)d\theta$

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Answer

$\frac{2}{5}\sqrt{\left(x+3\right)^{5}}-2\sqrt{\left(x+3\right)^{3}}+C_0$

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