Question

If \( f'(x) = \frac{x}{\sqrt{1 + x^2}} \) and \( f(0) = 0 \), then \( f(x) = \):

• A. \( \frac{2}{3}(1 + x^2)^{\frac{3}{2}} - 6(1 + x^2)^{1/2} \)
• B. \( \frac{2}{3}(1 + x^2)^{\frac{5}{2}} \)
• C. \( \frac{2}{3}(1 + x^2)^{\frac{3}{2}} \)
• D. \( \frac{2}{3}(1 + x^2)^{\frac{1}{2}} \)

Hint:

When integrating \( \frac{x}{\sqrt{1 + x^2}} \), use substitution to simplify the integral.

Answer 1

The Correct Option is B

Step 1: Integrate to find \( f(x) \). Integrating \( f'(x) = \frac{x}{\sqrt{1 + x^2}} \), we use substitution to find the solution.
Step 2: Conclusion. Thus, the value of \( f(x) \) is \( \frac{2}{3}(1 + x^2)^{\frac{5}{2}} \).