Indefinite Integral of ( \frac{1}{\sqrt{3 - 2x^2}} )

Overview

• The integral ( \int \frac{1}{\sqrt{3 - 2x^2}} , dx ) is not straightforward because no traditional method such as direct substitution applies.
• Use trigonometric substitution by leveraging trigonometric identities.

Trigonometric Identity

• Basic identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )
• By manipulation: ( \cos^2(\theta) = 1 - \sin^2(\theta) )

Steps for Solution

Factor and Rewrite

• Factor out 3 from the denominator:
  • ( 3 - 2x^2 = 3(1 - \frac{2}{3}x^2) )

Trigonometric Substitution

• Set ( \frac{2}{3}x^2 = \sin^2(\theta) )
• Solve for x and ( \theta ):
  • ( \sqrt{\frac{2}{3}}x = \sin(\theta) )
  • ( \theta = \arcsin\left(\sqrt{\frac{2}{3}}x\right) )
  • ( x = \sqrt{\frac{3}{2}} \sin(\theta) )

Differentiate for dx

• Find ( dx ) in terms of ( d\theta ):
  • ( \frac{dx}{d\theta} = \sqrt{\frac{3}{2}} \cos(\theta) )
  • ( dx = \sqrt{\frac{3}{2}} \cos(\theta) , d\theta )

Substitute into Integral

• Substitute ( x ) and ( dx ):
  • Replace dx: ( \sqrt{\frac{3}{2}} \cos(\theta) , d\theta )
  • Replace denominator: ( \sqrt{3 \cdot \cos^2(\theta)} )

Simplify and Integrate

• Cancel out terms: ( \cos(\theta) ) and ( \sqrt{3} )
• Integral simplifies to:
  • ( \int \frac{1}{\sqrt{2}} , d\theta )
  • Equals: ( \frac{1}{\sqrt{2}} (\theta + C) )

Reverse Substitution

• Substitute back ( \theta ) with x:
  • ( \theta = \arcsin\left(\sqrt{\frac{2}{3}}x\right) )
  • Final result: ( \frac{1}{\sqrt{2}} \arcsin\left(\sqrt{\frac{2}{3}}x\right) + C )

Conclusion

• The antiderivative of ( \frac{1}{\sqrt{3 - 2x^2}} ) is ( \frac{1}{\sqrt{2}} \arcsin\left(\sqrt{\frac{2}{3}}x\right) + C ).
• Understanding trigonometric identities and substitutions can simplify complex integrals.