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Step-by-step Solution

Integral of x(x+2)^0.5

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Answer

$\frac{2}{5}\sqrt{\left(x+2\right)^{5}}-\frac{4}{3}\sqrt{\left(x+2\right)^{3}}+C_0$

Step-by-step explanation

Problem to solve:

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

Solve the integral $\int x\sqrt{x+2}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=2\tan\left(\theta\right)^{2} \\ dx=4\sec\left(\theta\right)^2\tan\left(\theta\right)d\theta\end{matrix}$
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Substituting in the original integral, we get

$\int8\tan\left(\theta\right)^{2}\sqrt{2\tan\left(\theta\right)^{2}+2}\sec\left(\theta\right)^2\tan\left(\theta\right)d\theta$

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Answer

$\frac{2}{5}\sqrt{\left(x+2\right)^{5}}-\frac{4}{3}\sqrt{\left(x+2\right)^{3}}+C_0$