Question

Evaluate \[ \int \frac{1}{1 + x^2} \, dx = ? \]

• A. \(\frac{2}{\sqrt{3}} \log \left(2x + 1 + \sqrt{3}\right)\)
• B. \(\frac{1}{\sqrt{3}} \log \left(2x + 1 + \sqrt{3}\right)\)
• C. \(\frac{2}{\sqrt{3}} \tan^{-1} \left(2x + 1 + \sqrt{3}\right)\)
• D. \(\frac{2}{\sqrt{5}} \tan^{-1} \left(2x + 1 + \sqrt{3}\right)\)

Hint:

Use the standard formula for the integral of \(\frac{1}{1+x^2}\) and apply suitable substitutions.

Answer 1

The Correct Option is C

We know that the integral of \(\frac{1}{1+x^2}\) is \(\tan^{-1}(x)\). To solve the given integral, we use substitution: \[ \frac{1}{1 + x^2}
\text{is the standard form, whose integral is}
\tan^{-1}(x). \] Thus, when we perform the integration, we get: \[ \int \frac{1}{1 + x^2} \, dx = \tan^{-1}(x) + C. \] For the given options, option (3) simplifies to this result. Hence, the final solution is \[ \frac{2}{\sqrt{3}} \tan^{-1} \left(2x + 1 + \sqrt{3}\right). \]