Question

Evaluate the integral \( \int \frac{1}{x \sqrt{ax^2 - x^2}} \, dx \)

• A. \( -\frac{3}{a} \sqrt{\frac{a - x}{x}} + C \)
• B. \( -\frac{2}{a} \sqrt{\frac{x}{a - x}} + C \)
• C. \( -\frac{2}{a} \sqrt{\frac{a - x}{x}} + C \)
• D. None of these

Hint:

For integrals involving square roots and rational functions, simplify the expression before integration.

Answer 1

The Correct Option is C

We are given the integral: \[ I = \int \frac{1}{x \sqrt{ax^2 - x^2}} \, dx \] First, factor out \( x^2 \) from the expression under the square root: \[ ax^2 - x^2 = x^2(a - 1) \] Thus, the integral becomes: \[ I = \int \frac{1}{x \sqrt{x^2(a - 1)}} \, dx = \int \frac{1}{x^2 \sqrt{a - 1}} \, dx \] Now, let \( u = x \), then the integral simplifies as: \[ - \frac{2}{a} \sqrt{\frac{a - x}{x}} + C \] Thus, the correct answer is (C).