Question

Evaluate the integral \[ \int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \frac{1}{\left(x + \sqrt{1 - x^2}\right) \cdot \left(1 - x^2\right)} \, dx = \]

• A. \( \log(\sqrt{3} + 1) \)
• B. \( \log(\sqrt{3} - 1) \)
• C. \( \log(3 + \sqrt{3}) \)
• D. \( \log(3 - \sqrt{3}) \)

Hint:

Trigonometric substitution is very effective when dealing with square roots involving \(1 - x^2\).

Answer 1

The Correct Option is D

Use the substitution \( x = \sin \theta \). Then \( \sqrt{1 - x^2} = \cos \theta \), and the integrand simplifies significantly. Transform the limits accordingly and integrate the simplified expression. The result simplifies to \( \log(3 - \sqrt{3}) \).