Before You Begin

Question #40408

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_0^{2\pi} \frac{dx}{2 + \cos x + |\sin x|}$ . Which of the following inequalities is true?

Options

A.
$I \ge \sqrt{2}\pi$
B.
$\pi \le I < \sqrt{2}\pi$
C.
$\frac{3\pi}{4} \le I < \pi$
D.
$I < \frac{3\pi}{4}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Tangent Half-Angle Substitution|Inequalities in Definite Integrals|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle absolute values in trigonometric integrals and rigorously bound the result.

The primary trap lies in the Weierstrass substitution $t = \tan(x/2)$ .

If a student blindly applies this substitution over the entire interval $[0, 2\pi]$ , the term $|\sin x|$ transforms to $\frac{2|t|}{1+t^2}$ , creating a non-differentiable cusp at $t=0$ in the $t$ -domain.

Students commonly drop the absolute value, incorrectly integrating the smooth function $\frac{2}{t^2+2t+3}$ over $(-\infty, \infty)$ to get $\sqrt{2}\pi$ , leading them to choose the invalid upper bound (Option A).

Correct evaluation requires splitting the integral to respect the cusp, followed by precise trigonometric bounding using Taylor series properties ( $\arctan x < x$ ) to find the tight inequality.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Tangent Half-Angle Substitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free