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Step-by-step Solution

Integral of $\frac{x^2}{\sqrt{5-x^2}}$ with respect to x

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Answer

$\frac{5}{2}arcsin\left(\frac{x}{\sqrt{5}}\right)-\frac{1}{2}\sqrt{5-x^2}x+C_0$

Step-by-step explanation

Problem to solve:

$\int\:\frac{x^2}{\sqrt{5-x^2}}dx$
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Resolver la integral $\int\frac{x^2}{\sqrt{5-x^2}}dx$ mediante el método de sustitución trigonométrica. Tomamos el cambio de variable

$\begin{matrix}x=\sqrt{5}\sin\left(\theta\right) \\ dx=\sqrt{5}\cos\left(\theta\right)d\theta\end{matrix}$
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Sustituyendo en la integral original, obtenemos

$\int\frac{11.1803\sin\left(\theta\right)^2\cos\left(\theta\right)}{\sqrt{5-5\sin\left(\theta\right)^2}}d\theta$

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Answer

$\frac{5}{2}arcsin\left(\frac{x}{\sqrt{5}}\right)-\frac{1}{2}\sqrt{5-x^2}x+C_0$