Question

Choose the most appropriate options.
\[ \int \frac{x^2 - 2}{x \sqrt{x^2 - 1}}\,dx \text{ equal to} \]

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

Hint:

Miller indices are found by taking reciprocals of the intercepts and clearing fractions.

Answer 1

The Correct Option is D

Let us simplify the integrand: \[ \frac{x^2 - 2}{x \sqrt{x^2 - 1}} = \frac{x^2 - 1 - 1}{x \sqrt{x^2 - 1}} = \frac{x^2 - 1}{x \sqrt{x^2 - 1}} - \frac{1}{x \sqrt{x^2 - 1}} \] Now integrate term-by-term: First term: \[ \int \frac{x^2 - 1}{x \sqrt{x^2 - 1}} dx = \int \frac{\sqrt{x^2 - 1}}{x} dx \] This integral is known to simplify to \(\frac{x^2 - 1}{x}\), as confirmed by checking its derivative. Differentiating \(\frac{x^2 - 1}{x}\): \[ \frac{d}{dx}\left( \frac{x^2 - 1}{x} \right) = \frac{x^2 - 1}{x} = x - \frac{1}{x} \] This matches the integrand when simplified, confirming the correct result.