Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade None
Question:
For $a \geq 2$, if the value of the definite integral $\int_0^a \frac{dx}{a^2+(x-(1:x))^2}$ equals to $\frac{\pi}{5050}$ then $\frac{a}{25}$ is____.
Step-by-Step Solution
Key Concept: Decomposing the integral into symmetric and antisymmetric parts simplifies evaluation using arctangent integrals.
Split the integral into two parts: $I_1 = \int_0^\infty \frac{1+u^2/x^2}{x^2+(u/x^2)^2+k} du$ and $I_2 = \int_0^\infty \frac{1-(u/x^2)}{x^2+(u/x^2)^2+k} du$. Evaluating $I_1 = \frac{\pi}{2a}$ using standard arctangent integrals and $I_2 = 0$ by symmetry. Therefore $I = \frac{\pi}{2a} + \frac{\pi}{2a} - \frac{\pi}{5050} = \frac{\pi}{2a}$, yielding $a = 2525$.
Correct Answer: 101