Kerala KEAM 5 June 2024 Solved Paper

We are asked to evaluate the integral: $$I = \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx$$Step 1: Let $f(x) = \ln \left( \frac{1000 + x}{1000 - x} \right)$. Observe that the integrand is an odd function because: $$f(-x) = \ln \left( \frac{1000 - x}{1000 + x} \right) = -\ln \left( \frac{1000 + x}{1000 - x} \right) = -f(x)$$Step 2: The integral of an odd function over a symmetric interval $[-a, a]$ is 0, because the areas above and below the x-axis cancel each other out. Therefore: $$\int_{-500}^{500} f(x) \, dx = 0$$Thus, the value of the integral is $0$.Therefore, the correct answer is option (D).Quick Tip: When integrating odd functions over symmetric intervals, the integral is always zero because the positive and negative areas cancel out.