1 markMCQSection A

The value of $\int_{-1}^{1} \frac{x^3}{x^2+2|x|+1}\,dx$ is

Q: The value of $\int_{-1}^{1} \frac{x^3}{x^2+2|x|+1}\,dx$ is

**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.

Integrals

Definite Integrals — Odd/Even Function Property

Options

(A)0

(B)

\log 2

(C)

2\log 2

(D)

\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.

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

Marking Scheme

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

Analysis & Explanation

Concept: Property of definite integrals over symmetric limits.

Let $f(x) = \frac{x^3}{x^2+2|x|+1}$ . Since $x^2+2|x|+1 = (|x|+1)^2$ is an even function, and x^3 is odd, f(x) is an odd function: $f(-x) = -f(x)$ . 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)\ $\log 2$ and (D)\ $\frac{1}{2}\log 2$ would arise from confusing this with an integral of the logarithmic type, ignoring that the numerator here is an odd cubic, not linear.
(C)\ $2\log 2$ 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.

Common Mistakes

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

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 $x=0$ , yet they can still form part of perfectly well-behaved odd or even functions.

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Frequently Asked Questions

Why is the answer 0 without integrating?

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

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

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