Question

\( \int \frac{x}{(1-x^2)\sqrt{2 - x^2} dx = \)}

• A. \( \log \left| \frac{\sqrt{2 - x^2} + 1}{\sqrt{2 - x^2} - 1} \right| + c \)
• B. \( \frac{1}{2} \log \left| \frac{2 - x^2}{1 - x^2} \right| + c \)
• C. \( \frac{1}{2} \log \left| \frac{1 + \sqrt{2 - x^2}}{1 - \sqrt{2 - x^2}} \right| + c \)
• D. \( \log \left| \frac{1 - x^2}{\sqrt{2 - x^2}} \right| + c \)

Hint:

Use trigonometric substitution when square roots of quadratic expressions appear in integrals.

Answer 1

The Correct Option is C

Let \( I = \int \frac{x}{(1-x^2)\sqrt{2 - x^2}} dx \). Use substitution \( x = \sqrt{2} \sin \theta \Rightarrow dx = \sqrt{2} \cos \theta d\theta \). The expression simplifies using trigonometric identities leading to the standard result.