Let $f(x) = \int \frac{x^2\,dx}{(1+x^2)(1+\sqrt{1+x^2})}$ and $f(0)=0$. Then, $f(1)$ is
- A. $\log(1+\sqrt{2})$
- B. $\log(1+\sqrt{2}) - \frac{\pi}{4}$
- C. $\log(1+\sqrt{2}) + \frac{\pi}{4}$
- D. None of these
✅ Correct Answer: $\log(1+\sqrt{2}) - \frac{\pi}{4}$
Explanation
Integrate using trigonometric substitution $x=\tan\theta$, then simplify. The result gives $f(1)=\log(1+\sqrt{2}) - \frac{\pi}{4}$.
🎯 Happy Preparation — ACME Academy
