Question

Evaluate the integral: \( \int \frac{1}{x^3} \sqrt{1 - \frac{1}{x^2}} \text{dx} = \)

• A. \( \frac{-1}{6}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• B. \( \frac{1}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• C. \( \frac{-1}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• D. \( \frac{4}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• E. \( \frac{-4}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)

Hint:

Always look for a function and its derivative when dealing with complex integrals. Factoring out terms or rewriting fractions with negative exponents (like $1/x^2$ as $x^{-2}$) often reveals a clear substitution path.

Answer 1

The Correct Option is B