Question #41465

This question tests the student's vigilance regarding the domain of integration and hidden singularities.

A common trap in JEE Advanced is applying the Newton-Leibniz formula (or Integration by Parts) blindly across an interval where the integrand or its antiderivative is undefined.

Here, the integrand has a vertical asymptote at $x = \pi/2$ because $\ln(\cos^2 x)$ is undefined and $\cos^3 x = 0$.

If a student ignores this domain restriction and blindly applies IBP, they will find an antiderivative $F(x) = 2\sec x \ln(\cos^2 x) + 4\sec x$ and incorrectly evaluate $F(\pi) - F(0) = -8$.

Recognizing the singularity and testing for convergence reveals that the improper integral diverges to $-\infty$.