Inverse trig substitution integral

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SUMMARY

The discussion focuses on the integration of the function \(\int \frac{x^2}{\sqrt{9-x^2}} \, dx\) using the inverse trigonometric substitution method. The substitution \(x = 3\sin\theta\) is applied, leading to the transformation of the integral into \(\int 9\sin^2\theta \, d\theta\). Participants suggest simplifying the integrand before proceeding with integration techniques, emphasizing the importance of trigonometric identities in this context.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with trigonometric identities
  • Knowledge of substitution methods in integration
  • Basic skills in manipulating algebraic expressions
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  • Study the application of trigonometric identities in integration
  • Learn about the integration of \(\sin^2\theta\) using reduction formulas
  • Explore advanced techniques in integration by parts
  • Review the concept of inverse trigonometric functions and their derivatives
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Students and educators in calculus, mathematicians focusing on integration techniques, and anyone seeking to enhance their skills in solving integrals involving trigonometric substitutions.

James889
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Hi,

I need to integrate the following:

[tex]\int \frac{x^2}{\sqrt{9-x^2}}[/tex]

So let [tex]x = 3sin\theta[/tex]
[tex]\frac{dx}{d\theta} = 3cos\theta[/tex]

So i now have the integral of [tex]\frac{9sin^2\theta \cdot 3cos\theta}{3cos\theta}[/tex]

How do i go about the integration from here? parts?
 
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Try simplifying the integrand first.
 

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