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Question #40288

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the definite integral: $I = \int_{0}^{3/2} \left( \frac{1}{2x^2 - 3x + 1} - \frac{4x^2 - 3x}{(2x^2 - 3x + 1)^2} \right) dx$

Options

A.
$\frac{3}{2}$
B.
$-\frac{3}{2}$
C.
$0$
D.
Does not exist

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

Well-definedness|Partial Fractions Decomposition|Integrating f(x) + xf'(x) forms|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the 'Denominator Restriction' trap in definite integrals.

Students often mechanically apply the identity $\int (f(x) + xf'(x)) dx = xf(x)$ and evaluate it at the boundaries, completely ignoring the well-definedness of the integrand.

Because the rational function has roots in its denominator at $x=1/2$ and $x=1$ (both inside the interval $[0, 3/2]$ ), the Fundamental Theorem of Calculus cannot be applied directly across these non-integrable singularities.

The integral is improper and strictly diverges, making the algebraic antiderivative evaluation a lethal collision.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Well-definedness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free