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Question #41442

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the definite integral: $I = \int_0^{\frac{21\pi}{2}} x \sqrt{1-\cos 2x} \cos^4 x \, dx$

Options

A.
$\sqrt{2} \left( 22\pi + \frac{8}{75} \right)$
B.
$\sqrt{2} \left( \frac{441\pi}{20} \right)$
C.
$\sqrt{2} \left( 20\pi + \frac{8}{75} \right)$
D.
$\sqrt{2} \left( 22\pi - \frac{8}{75} \right)$

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

Reduction Formulae|Definite Integral Properties (King's Property)|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question engineers a lethal collision between King's Property and piecewise periodic reduction.

Students frequently memorize the shortcut $\int_0^a x f(x) dx = \frac{a}{2} \int_0^a f(x) dx$ and blindly apply it to the entire interval $[0, \frac{21\pi}{2}]$ .

However, this property strictly requires $f(a-x) = f(x)$ , which fails here because the upper limit is not an integer multiple of the period $\pi$ .

This forces a piecewise split.

A secondary trap lies in the substitution $x = 10\pi + t$ for the fractional interval, where students often forget to distribute the $10\pi$ shift, leading to the incorrect option C.

Quick Facts About This Question

Question Information

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