Question

If \( \int \left( x \sin^{-1} x + \sin^{-1} x (1 - x^2)^{3/2} + \frac{x}{1 - x^2} \right) dx = g(x) + C \), where C is the constant of integration, then \( g\left(\frac{1}{2}\right) \) equals:

• A. \( \frac{\pi}{6} \sqrt{3} \)
• B. \( \frac{\pi}{4} \sqrt{2} \)
• C. \( \frac{\pi}{4} \sqrt{3} \)
• D. \( \frac{\pi}{6} \sqrt{2} \)

Hint:

When integrating complex expressions involving inverse trigonometric functions: - Apply standard integration formulas for inverse sine and cosine functions. - Recognize patterns in integrals and use substitution to simplify the integral before solving.

Answer 1

The Correct Option is C

We begin by evaluating the given integral. Use standard integral formulas and apply limits as required to evaluate \( g\left( \frac{1}{2} \right) \). By carefully solving the integral, we find: \[ g\left( \frac{1}{2} \right) = \frac{\pi}{4} \sqrt{3}. \]