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GATE 2023 CE (CE1) – Question 11

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 1 mark · Multiple choice

For the integral $I = \int_{-1}^{1}\frac{1}{x^2}dx$ which of the following statements is TRUE?

  1. $I = 0$
  2. $I = 2$
  3. $I = -2$
  4. The integral does not converge

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Correct answer: (D) The integral does not converge

Explanation

The function $\frac{1}{x^2}$ is not bounded at $x = 0$, and $\int_0^1\frac{dx}{x^2}$ diverges (it is $\left[-\frac{1}{x}\right]$ and goes to infinity at 0). So the integral does not converge. The value $-2$ from the antiderivative is wrong because it ignores the singularity.