Integrate x^2/sqrt(4-x^2): Solution Steps

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SUMMARY

The integral of the function \(\frac{x^2}{\sqrt{4-x^2}}\) can be effectively solved using trigonometric substitution. Specifically, substituting \(x = 2 \sin(\theta)\) simplifies the integral significantly. This method transforms the integral into a more manageable form, allowing for straightforward integration. The discussion confirms that integration by parts is not the optimal approach for this particular integral.

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  • Understanding of trigonometric identities and substitutions
  • Familiarity with integral calculus and techniques
  • Knowledge of the properties of square roots in integrals
  • Experience with integration by parts
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Students and educators in calculus, mathematicians focusing on integration techniques, and anyone seeking to deepen their understanding of trigonometric substitutions in integral calculus.

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<br /> \int{ \frac{x^2}{\sqrt{4-x^2}} }<br />

How would I do that?
I tried integrating by parts, letting u = x^2 and v' = 1/sqrt(4-x^2) and many other combinations, but I just can't seem to get the result.
 
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A trig substitution will work, surely.
 
Let x= 2 sin(\theta)
 

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