Question

Evaluate the integral: $$\int \sin^{-1} \left( \sqrt{\frac{x - a}{x}} \right) dx$$

• A. $x \cos^{-1} \left(\sqrt{\frac{a}{x}}\right) - \sqrt{ax - a^2} + C$
• B. $x \sec^{-1} \left(\sqrt{\frac{a}{x}}\right) + \sqrt{x^2 - ax} + C$
• C. $x \sin^{-1} \left(\sqrt{\frac{x}{a}}\right) + \sqrt{x^2 + ax} + C$
• D. $\frac{x}{a} \sin^{-1} \left(\frac{x}{a}\right) + \frac{x^2}{a} \sqrt{1 + a^2} + C$

Hint:

For inverse trigonometric integrals, use suitable trigonometric identities and substitutions to simplify expressions.

Answer 1

The Correct Option is A

Step 1: Use the substitution Let $$t = \sin^{-1} \left( \sqrt{\frac{x - a}{x}} \right).$$ Then, $$\sin t = \sqrt{\frac{x - a}{x}}.$$ Squaring both sides, $$\sin^2 t = \frac{x - a}{x}.$$ Thus, $$x \sin^2 t = x - a.$$ Step 2: Differentiate both sides Using implicit differentiation, $$2x \sin t \cos t \frac{dt}{dx} + \sin^2 t = 1.$$ Rearranging, $$\frac{dt}{dx} = \frac{1 - \sin^2 t}{2x \sin t \cos t}.$$ Using trigonometric identities, solve the integral, leading to: $$x \cos^{-1} \left(\sqrt{\frac{a}{x}}\right) - \sqrt{ax - a^2} + C.$$