MHB Is Substitution Effective for Integrating \(\frac{x^2}{(4-x^2)^{3/2}} \, dx\)?

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The integral \(\frac{x^2}{(4 - x^2)^{3/2}} \, dx\) can be evaluated using the substitution \(x = 2\sin{(\theta)}\), which leads to \(dx = 2\cos{(\theta)}\,d\theta\). This substitution simplifies the expression significantly, allowing for easier integration. Participants express confusion about the integral's complexity but agree that the trigonometric substitution is a promising approach. The discussion emphasizes the effectiveness of substitution in solving this integral. Ultimately, the substitution method is highlighted as a viable solution strategy.
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Evaluate the Integral.

Please help! I am so lost on this one!

$$\frac{x^2}{(4 - x^2)^\frac{3}{2}} \, dx$$
 
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shamieh said:
Evaluate the Integral.

Please help! I am so lost on this one!

$$\frac{x^2}{(4 - x^2)^\frac{3}{2}} \, dx$$

The substitution \displaystyle \begin{align*} x = 2\sin{(\theta)} \implies dx = 2\cos{(\theta)}\,d\theta \end{align*} will probably work...
 
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