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Question #40436

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral: $I = \int_0^\pi \frac{|\sin 2x| + 8\cos^2 x}{5 + 4\cos 2x} dx$

Options

A.
$\frac{2\pi}{3} + \frac{1}{2} \ln 3$
B.
$\pi + \frac{1}{2} \ln 3$
C.
$\frac{1}{2} \ln 3$
D.
$\frac{4\pi}{3} + \frac{1}{2} \ln 3$

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

Double and Triple Angle Ratios|Logarithmic Integration via Trig Manipulation|Substitution (Change of Variable)

Why This Question Is Tricky

Common Mistakes

This question tests the integration of complex trigonometric rational functions using double angle identities and logarithmic integration.

The core trap lies in the Weierstrass-style substitution ( $t = \tan x$ ) for the integral $\int_0^\pi \frac{dx}{5+4\cos 2x}$ .

Students often blindly apply the substitution over the entire interval $[0, \pi]$ , which yields bounds from $0$ to $0$ , incorrectly evaluating the rational part to $0$ (leading to Option C) or missing the reduction entirely (Option B).

The rigorous approach requires recognizing the discontinuity of $\tan x$ at $x = \pi/2$ and splitting the integral or using symmetry before substitution to account for the singularity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Double and Triple Angle Ratios
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free