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Question #41498

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral: $I = \int_0^{2\pi} \frac{2 \sin^2\left(\frac{x}{2}\right)}{(2+\cos x)(3+\cos x)} \, dx$

Options

A.
$2\pi(\sqrt{3} - \sqrt{2})$
B.
0
C.
$\pi(\sqrt{3} - \sqrt{2})$
D.
$2\pi(\sqrt{3} + \sqrt{2})$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Partial Fractions Decomposition|Tangent Half-Angle Substitution|The Quadrant Adjustment Logic

Why This Question Is Tricky

Common Mistakes

This question tests the 'Quadrant Adjustment Logic' trap inherent in the Weierstrass substitution ( $t = \tan(x/2)$ ).

Because $\tan(x/2)$ has a discontinuity at $x = \pi$ , blindly applying the substitution over the interval $[0, 2\pi]$ yields bounds from $t(0)=0$ to $t(2\pi)=0$ , incorrectly resulting in an integral of 0 (Option B).

To correctly evaluate it, students must either split the integral at the discontinuity or use the symmetry property $\int_0^{2a} f(x)dx = 2\int_0^a f(x)dx$ before substituting.

Furthermore, the resulting rational function requires careful Partial Fraction Decomposition, testing algebraic precision alongside advanced calculus concepts.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Partial Fractions Decomposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free