Question

Evaluate the integral \[ \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx \]

• A. 1000
• B. \( \ln 1000 \)
• C. \( \ln 500 \)
• D. 0
• E. \( \frac{1}{1000} \)

Hint:

When integrating odd functions over symmetric intervals, the integral is always zero because the positive and negative areas cancel out.

Answer 1

The Correct Option is D

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).