Question

If \[ \int \frac{dx}{(x \tan x + 1)^2} = f(x) + c, \] then \(\lim_{x \to \frac{\pi}{2}} f(x)\) is?

• A. \(\frac{\pi}{2}\)
• B. \(\frac{2}{\pi}\)
• C. \(\frac{1}{\pi}\)
• D. \(\infty\)

Hint:

Be cautious of integrals involving trigonometric functions at points where they become undefined or approach infinity.

Answer 1

The Correct Option is D

The given integral involves \(x \tan x\), and we need to evaluate its limit as \(x \to \frac{\pi}{2}\). Let's analyze the function and the behavior of \(\tan x\) near \(x = \frac{\pi}{2}\). As \(x \to \frac{\pi}{2}\), \(\tan x\) approaches infinity, which leads to the denominator becoming zero, and thus the function blows up. Therefore, the limit of \(f(x)\) as \(x \to \frac{\pi}{2}\) is \(\infty\). Hence, the correct answer is \(\infty\).