Question

\[ \int \frac{dx}{x\sqrt{x^2+4}}= \]

• A. \[ \frac12\log\left| \frac{\sqrt{x^2+4}+2}{\sqrt{x^2+4}-2} \right|+C \]
• B. \[ \frac14\log\left| \frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2} \right|+C \]
• C. \[ \frac12\log\left| \frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2} \right|+C \]
• D. \[ \frac14\log\left| \frac{\sqrt{x^2+4}+2}{\sqrt{x^2+4}-2} \right|+C \]

Hint:

Whenever radicals like $\sqrt{x^2+a^2}$ appear, try substituting the entire radical as a new variable. It often converts the integral into a standard rational form.

Answer 1

The Correct Option is B