---
title: "Integral of 1/x² from −1 to 1"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:integral-1-over-x2-minus1-to-1"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/integral-1-over-x2-minus1-to-1
site: BCT Engineering
---

# Integral of 1/x² from −1 to 1

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2074 Chaitra, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/integral-1-over-x2-minus1-to-1
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Evaluate the integral ∫_{−1}^{1} (1/x²) dx.

## Short answer

The integrand 1/x² is unbounded at x=0, making the integral improper. Splitting at 0 and evaluating the limit yields +∞, so the integral diverges.

## Answer

**The integral diverges.**

The integrand 1/x² is not defined at x = 0, which lies inside [−1, 1], and it is unbounded there. So this is an improper integral and must be split at 0:

∫_(−1)^1 dx/x² = ∫_(−1)^0 dx/x² + ∫₀¹ dx/x²

Consider the second part:

```text
∫₀¹ dx/x² = lim_(ε→0⁺) ∫_ε¹ x⁻² dx
          = lim [−1/x]_ε¹
          = lim (−1 + 1/ε)
          = +∞
```
Since one part diverges (the other part also diverges to +∞ by symmetry), the integral does not exist as a finite number.

**Warning:** applying the fundamental theorem blindly gives [−1/x]_(−1)^1 = −1 − 1 = −2, which is wrong. The integrand is positive, so a negative answer is impossible, and the theorem requires the integrand to be continuous on the whole interval.

**Answer: the integral diverges (equals +∞).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The integral is improper because
   A. the limits are infinite
   B. 1/x^2 is unbounded at x = 0, inside [-1,1]
   C. 1/x^2 is negative
   D. the integral is indefinite
   Answer: B. The singularity lies in the interval. (Key point: Why improper)

2. The correct approach is to
   A. apply the fundamental theorem directly
   B. substitute x = -t
   C. split at 0 and use limits as ε → 0+
   D. use symmetry to double the integral from -1 to 0
   Answer: C. Each part is evaluated as a limit. (Key point: Splitting)

3. Applying the fundamental theorem blindly gives -2, which is wrong because
   A. it should be +2
   B. it should be 0
   C. 1/x^2 is odd
   D. the integrand is positive and not continuous on the interval
   Answer: D. A positive integrand cannot have a negative integral. (Key point: Blind result)

4. The integral
   A. diverges to +∞
   B. equals -2
   C. equals 2
   D. equals 0
   Answer: A. The part from 0 to 1 gives lim(-1 + 1/ε) = ∞. (Key point: Answer)

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