Question

The value of $ \int \frac{dx}{\sqrt{2ax - x^2}} $

• A. \( \sin^{-1}(x - 1) + C \)
• B. \( \sin^{-1}(2x - 1) + C \)
• C. \( \sin^{-1}(x + 1) + C \)
• D. \( -\sqrt{2ax - x^2} + C \)

Hint:

When integrating expressions involving square roots of quadratic terms, completing the square can often simplify the expression into a standard form.

Answer 1

The Correct Option is A

We need to evaluate the integral: \[ I = \int \frac{dx}{\sqrt{2ax - x^2}}. \] To solve this, let's first complete the square in the expression under the square root. We have: \[ 2ax - x^2 = -(x^2 - 2ax) = -(x^2 - 2ax + a^2 - a^2) = -\left[(x - a)^2 - a^2\right]. \] Thus, the integral becomes: \[ I = \int \frac{dx}{\sqrt{a^2 - (x - a)^2}}. \] This is a standard form for the integral of the type \( \int \frac{dx}{\sqrt{A^2 - x^2}} \), which is known to have the solution: \[ \sin^{-1}\left(\frac{x - a}{a}\right) + C. \] Thus, the correct answer is: \[ I = \sin^{-1}(x - 1) + C. \] Thus, the correct answer is (A).