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Question #41505

Difficulty: 8/10 | 🟢 Free

The Question

Let $I(x) = \int \frac{2x-6}{x^4 - 12x^3 + 61x^2 - 150x + 25m - m^2} dx$ . If $I(x)$ can be expressed in the form $\frac{1}{25-2m} \ln \left( \frac{g(x)}{h(x)} \right) + C$ for all $x \in \mathbb{R}$ , where $g(x)$ and $h(x)$ are monic quadratic polynomials and $C$ is a constant of integration, find the sum of all possible integer values of $m$ .

Options

A.
75
B.
100
C.
33
D.
50

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

Completing the Square in Quadratics|Partial Fractions Decomposition|Logarithmic Differentiation Domain

Why This Question Is Tricky

Common Mistakes

This question tests the integration of rational functions using partial fractions, combined with a subtle domain trap involving logarithmic functions.

Students often mechanically integrate to find $\ln|f(x)|$ but fail to recognize that removing the absolute value sign for all $x \in \mathbb{R}$ requires the argument to be strictly positive.

This forces the resulting monic quadratic polynomials to have negative discriminants, linking integral calculus with algebraic invariants and coordinate geometry constraints.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Completing the Square in Quadratics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free