Question 91 markMCQSection A

The value of ∫−11x3x2+2∣x∣+1 dx\int_{-1}^{1} \frac{x^3}{x^2+2|x|+1}\,dx is

Integrals
Definite Integrals — Odd/Even Function Property

Options

(A)0
(B)log⁡2\log 2
(C)2log⁡22\log 2
(D)12log⁡2\frac{1}{2}\log 2

Official answer

Correct option: (A) 0


The integrand is an odd function on a symmetric interval, so the definite integral evaluates to zero without any further computation.

Answer keywords:odd functiondefinite integralsymmetric limitseven function propertyintegral propertyx^2+2|x|+1

Marking scheme

  1. 1.1 mark: awarded only for selecting option (A) 0; no partial credit in MCQs.

Hint

Check whether the integrand is odd or even before integrating — symmetric limits with an odd integrand always give 0.

Quick oral answer

Since x2+2∣x∣+1=(∣x∣+1)2x^2+2|x|+1 = (|x|+1)^2 is even and x^3 is odd, the whole integrand is odd, so its integral over [−1,1][-1,1] is zero by the odd-function symmetry property.

Common mistakes

  1. 1.Splitting the integral into [−1,0][-1,0] and [0,1][0,1] and attempting full log-based integration instead of first checking odd/even symmetry, wasting exam time.
  2. 2.Misjudging x3x2+2∣x∣+1\frac{x^3}{x^2+2|x|+1} as an even function because of the |x| term, when the numerator's odd cube dominates the parity.
  3. 3.Forgetting that (∣x∣+1)2(|x|+1)^2 is always even, which is key to correctly identifying the whole expression as odd.

Analysis and explanation

Concept: Property of definite integrals over symmetric limits.


Let f(x)=x3x2+2∣x∣+1f(x) = \frac{x^3}{x^2+2|x|+1}. Since x2+2∣x∣+1=(∣x∣+1)2x^2+2|x|+1 = (|x|+1)^2 is an even function, and x^3 is odd, f(x) is an odd function: f(−x)=−f(x)f(-x) = -f(x). By the standard property, the integral from -a to a of an odd function is always 0, regardless of its exact algebraic form.


Why the distractors fail:

  • (B)\ log⁡2\log 2 and (D)\ 12log⁡2\frac{1}{2}\log 2 would arise from confusing this with an integral of the logarithmic type, ignoring that the numerator here is an odd cubic, not linear.
  • (C)\ 2log⁡22\log 2 doubles a nonzero logarithmic value that does not apply here since the true value is 0 by symmetry, not by evaluating antiderivatives.
  • All three distractors ignore the odd-function shortcut and instead assume the integral requires splitting and log-based antiderivatives, leading to non-zero (and incorrect) answers.

Exam tip: Always check the odd/even nature of the integrand before attempting substitution — it can save significant time in a 1-mark MCQ.

Interesting facts

The property that the integral from -a to a of an odd function equals 0 is one of the most frequently tested shortcuts in CBSE definite integral MCQs, appearing almost every year.

Functions involving |x| are continuous everywhere but not differentiable at x=0x=0, yet they can still form part of perfectly well-behaved odd or even functions.

Question details

Q9 of 44 questions

Board
CBSE
Class
Class 12
Subject
Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Integrals
Topic
Definite Integrals — Odd/Even Function Property
Bloom's level
apply
Syllabus topic
Unit 4: Integrals — Properties of Definite Integrals (odd and even functions)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why is the answer 0 without integrating?

Because the integrand x3x2+2∣x∣+1\frac{x^3}{x^2+2|x|+1} is an odd function (f(−x)=−f(x)f(-x) = -f(x)), and the definite integral of any odd function over a symmetric interval [−a,a][-a, a] is always 0.

Is x2+2∣x∣+1x^2+2|x|+1 an odd or even function?

It is even, since replacing x by -x leaves |x| unchanged, i.e. (∣−x∣+1)2=(∣x∣+1)2(|-x|+1)^2 = (|x|+1)^2. Combined with the odd numerator x^3, the overall integrand becomes odd.