Indefinite Integration
Integral Calculus-1
star_batch_jee_advanced_2025
Grade None
Question:
If $f'(x) = \frac{1}{-x + \sqrt{x^2 + 1}}$ and $f(0) = -\frac{1 + \sqrt{2}}{2}$, then the value of $f(5)$ will be :
More than 12
Less than 5
More than 5
None of these
Step-by-Step Solution
Key Concept: The boundary condition at $x = 0$ directly determines the constant of integration.
Computing $f(x) = \int\left(x + \sqrt{x^2 + 1}\right) dx = \frac{x^2}{2} + \frac{x\sqrt{x^2+1}}{2} + \frac{1}{2}\log\left|x + \sqrt{x^2+1}\right| + c$. Using the boundary condition $f(0) = c$ and the given value $f(0) = -\frac{1+\sqrt{2}}{2}$, the constant of integration is determined.
Correct Answer: 1,3