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Question #41419

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral $I = \int_0^{3\pi} \frac{1 + 2\sin x - \cos x}{5 + 3\cos x} \, dx$

Options

A.
$\pi + \frac{4}{3}\ln 2$
B.
$\frac{\pi}{3} + \frac{4}{3}\ln 2$
C.
$\frac{2\pi}{3}$
D.
$\pi - \frac{4}{3}\ln 2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

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

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Tangent Half-Angle (Weierstrass) substitution combined with partial fraction decomposition.

The lethal trap lies in the 'Quadrant Adjustment Logic' due to the jump discontinuity of $t = \tan(x/2)$ at $x = \pi$ .

Students who blindly evaluate the antiderivative from the boundary conditions $t(0)=0$ to $t(3\pi)=\infty$ without splitting the integral at the asymptote will completely miss the $\frac{2\pi}{3}$ contribution from the second interval, leading them directly to the trap option.

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