Question #40302

This question tests the student's understanding of improper integrals and the limitations of definite integral properties (specifically King's Property).

A common trap in JEE Advanced is to blindly apply the identity $\int_{-a}^{a} \frac{f(x)}{1+e^x} dx = \int_{0}^{a} f(x) \frac{1-e^x}{1+e^x} dx$ for an odd function $f(x)$.

While this identity holds for convergent integrals, applying it here artificially increases the degree of the numerator (since $1-e^x \sim -x$), masking the true singularity at $x=0$.

Students falling for this 'Modulus/Symmetry Trap' will incorrectly conclude the integral converges for $\alpha > -1$ because the Cauchy Principal Value exists.

However, true convergence of an improper integral over $[-a, a]$ requires independent convergence on $[-a, 0)$ and $(0, a]$, which reveals the correct domain $\alpha > 0$.