Evaluate the following indefinite integral

Question

Evaluate the following indefinite integral:

 

∫1/(x²√1-x²) dx

Answer 1

Trigonometric substitution: x = sin θ, dx = cos θ dθ

 

∫ 1/(x²√(1 - x²)) dx

 

= ∫ [1/(sin² θ√(1 - sin² θ))] cos θ dθ

 

= ∫ (cos θ) / (sin² θ√(cos² θ)) dθ

 

= ∫ csc² θ dθ

 

= -cot θ + C

 

= - (cos θ) / sin θ + C

 

= -√(1 - sin² θ) / sin θ + C

 

= -√(1 - x²) / x + C