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Question #41408

Difficulty: 8/10 | 🟢 Free

The Question

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

Options

A.
$2\sqrt{3} - \frac{2\pi}{3} - 2$
B.
$0$
C.
$2\sqrt{3} + \frac{2\pi}{3} - 2$
D.
$\sqrt{3} - \frac{\pi}{3} - 1$

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

Substitution (Change of Variable), The Sum-Difference Formula Constraints, Extraneous Solution Rejection, Chain Rule Inner Function Differentiability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle domain constraints during trigonometric substitution.

A common trap is blindly applying the identities $\arcsin(\frac{2x}{1+x^2}) = 2\arctan x$ and $\arccos(\frac{1-x^2}{1+x^2}) = 2\arctan x$ across the entire domain.

This ignores the principal value branch of the inverse sine function, which shifts when $x > 1$ (or $2\theta > \pi/2$ ).

Students falling for this trap will incorrectly conclude the integrand is identically zero, leading to option B.

The problem enforces 'Extraneous Solution Rejection' by requiring the integral to be split at the point of discontinuity of the inner function's derivative.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Substitution (Change of Variable)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free