Question

Evaluate the integral: \[ \int \sqrt{x + \sqrt{x^2 + 2}} \, dx. \]

• A. \(\frac{2}{3} (x + \sqrt{x^2 + 2})^{3/2} - 2(x + \sqrt{x^2 + 2})^{1/2} + C\)
• B. \(\frac{1}{3} (x + \sqrt{x^2 + 2})^{3/2} - 2(x + \sqrt{x^2 + 2})^{1/2} + C\)
• C. \( (x + \sqrt{x^2 + 2})^{-3/2} - 2(x + \sqrt{x^2 + 2})^{1/2} + C\)
• D. \(\frac{(x+\sqrt{x^2+2})^2 - 6}{3\sqrt{x+\sqrt{x^2+2}}} + C\)

Hint:

Use substitution to simplify complicated square roots and reduce the integrand to a manageable form.