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Question #41439

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a continuous function defined on $[-1, 1]$ which satisfies the differential equation $x(x^2 + y^2) dx + y(x^2 - y^2) dy = 0$ with the initial condition $f(0) = 1$ . Evaluate the definite integral: $I = \int_{-1}^{1} \left( x^2 \ln\left( \frac{f(x) - x}{f(x) + x} \right) + \frac{3x^2 \left( (f(x))^2 - x^2 \right)}{\sqrt{2x^4+1}} \right) dx$

Options

A.
0
B.
1
C.
2
D.
3

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

Even and Odd Function Symmetry|Homogeneous Function Homogeneity|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate branching logic during the implicit-to-explicit conversion of a differential equation's solution, and subsequently apply symmetry properties to evaluate a complex definite integral.

The primary trap lies in solving the quadratic in $y^2$ ; students often blindly take roots without checking domain constraints or initial conditions, leading to incorrect signs (e.g., $y^2 - x^2 = -\sqrt{2x^4+1}$ ).

Failing to select the correct positive branch prevents the simplification of the second integral term.

Furthermore, students must recognize that the explicit function $f(x)$ is even, which is required to deduce that the logarithmic term forms an odd function whose integral vanishes over the symmetric interval $[-1, 1]$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Even and Odd Function Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free