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Question #41456

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_0^{\frac{5\pi}{2}} \frac{dx}{13 - 4\cos x + 3\sin x}$ . Which of the following intervals contains the value of $I$ ?

Options

A.
$\left(0, \frac{\pi}{8}\right]$
B.
$\left(\frac{\pi}{8}, \frac{\pi}{4}\right]$
C.
$\left(\frac{\pi}{4}, \frac{3\pi}{8}\right]$
D.
$\left(\frac{3\pi}{8}, \frac{\pi}{2}\right]$

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

Inequalities in Definite Integrals|Tangent Half-Angle Substitution|The Sum-Difference Formula Constraints

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding domain constraints and discontinuities during integration by substitution.

The Weierstrass substitution $t = \tan(x/2)$ is a standard tool for rational trigonometric integrals, but it introduces a singularity at $x = \pi$ .

A common trap is to blindly evaluate the antiderivative at the endpoints $t(5\pi/2) = 1$ and $t(0) = 0$ , yielding the extraneous value $\pi/24$ , which perfectly matches the trap option A.

By forcing the integral across the branch cut, the student must split the integral and correctly apply the limits to $\pm\infty$ , revealing the hidden $\pi$ term.

The final step acts as a secondary gate, requiring the inverse trigonometric sum-difference formula to resolve the exact value.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Inequalities in Definite Integrals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free