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Question #41520

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the definite integral: $I = \int_{0}^{\pi} \frac{\sin^4 x}{(1-\cos x)^3} dx$

Options

A.
$-3\pi$
B.
$3\pi$
C.
0
D.
\text{Diverges}

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

Wallis' Formula|Tangent Half-Angle Substitution|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding improper integrals and singularities hidden by trigonometric substitutions.

The Weierstrass substitution $t = \tan(x/2)$ transforms the integrand into a rational function with an even-multiplicity root at $t=0$ ( $t^2$ in the denominator).

Students often blindly apply Wallis' formula (via $t = \tan\theta$ ) or partial fractions, evaluating the antiderivative at the boundaries while ignoring the infinite discontinuity at the lower limit.

Because the Gamma function extension of Wallis' formula (or algebraic simplification followed by ignoring the divergent $\csc^2(x/2)$ term) yields a finite negative value ( $-3\pi$ ) for this divergent integral, it acts as a 'false friend' trap, heavily penalizing rote memorization over analytical rigor.

Quick Facts About This Question

Question Information

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