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Question #41151

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral $I = \int_{-6}^{6} \frac{x^5 \ln(1+x^2) + [x]^2 + [-x]^2 + \{x\} + \{-x\}}{1 + 3^{[x] + [-x]}} \, dx$ where $[t]$ and $\{t\}$ denote the greatest integer and fractional part functions of $t$ , respectively.

Options

A.
114
B.
152
C.
228
D.
304

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

Greatest Integer Function Traps|Linearity and Partitioning|Even and Odd Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle definite integrals involving the greatest integer and fractional part functions over symmetric intervals.

The primary trap lies in the properties $[x] + [-x] = -1$ and {x} + {-x} = 1 for non-integers.

Students often incorrectly assume these evaluate to 0 everywhere or fail to recognize that the integer points form a set of measure zero, which does not affect the Riemann integral.

Furthermore, it requires identifying the hidden even symmetry of $[x]^2 + [-x]^2$ to correctly partition and sum the integral over unit intervals, punishing those who attempt brute-force integration without leveraging parity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Greatest Integer Function Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free