The integral of $\frac{1}{x^2}$ over the interval $[-1, 1]$ is improper because the integrand $\frac{1}{x^2}$ becomes undefined at $x = 0$. We need to split the integral into two parts and check whether each part converges:

$\int_{-1}^{1} \frac{1}{x^2} \, dx = \int_{-1}^{0} \frac{1}{x^2} \, dx + \int_{0}^{1} \frac{1}{x^2} \, dx.$

1. Integral $\int_{a}^{b} \frac{1}{x^2} \, dx$:

The antiderivative of $\frac{1}{x^2}$ is: $\int \frac{1}{x^2} \, dx = -\frac{1}{x}.$

2. Applying limits:

For the interval $[-1, 1]$, splitting the integral around the singularity at $x = 0$, we write: $\int_{-1}^{1} \frac{1}{x^2} \, dx = \lim_{\epsilon \to 0^+} \left( \int_{-1}^{-\epsilon} \frac{1}{x^2} \, dx + \int_{\epsilon}^{1} \frac{1}{x^2} \, dx \right).$

Evaluating each part:

• For $\int_{-1}^{-\epsilon} \frac{1}{x^2} \, dx$: $\int_{-1}^{-\epsilon} \frac{1}{x^2} \, dx = \left[-\frac{1}{x}\right]_{-1}^{-\epsilon} = \left(-\frac{1}{-\epsilon}\right) - \left(-\frac{1}{-1}\right) = \frac{1}{\epsilon} - 1.$

• For $\int_{\epsilon}^{1} \frac{1}{x^2} \, dx$: $\int_{\epsilon}^{1} \frac{1}{x^2} \, dx = \left[-\frac{1}{x}\right]_{\epsilon}^{1} = \left(-\frac{1}{1}\right) - \left(-\frac{1}{\epsilon}\right) = -1 + \frac{1}{\epsilon}.$

Adding the results:

$\lim_{\epsilon \to 0^+} \left( \frac{1}{\epsilon} - 1 + (-1 + \frac{1}{\epsilon}) \right) = \lim_{\epsilon \to 0^+} \left( \frac{2}{\epsilon} - 2 \right).$

As $\epsilon \to 0^+$, the term $\frac{2}{\epsilon} \to \infty$, so the integral diverges.

Conclusion:

The integral $\int_{-1}^{1} \frac{1}{x^2} \, dx$ does not converge because the function $\frac{1}{x^2}$ has a singularity at $x = 0$, causing the integral to diverge to infinity.

Would you like further clarification or visualizations for this?

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1. What is an improper integral?
2. How can we evaluate improper integrals with infinite limits?
3. What conditions cause an integral to diverge?
4. Can an integral diverge on only part of its domain?
5. What are other examples of integrals with singularities?

Tip: Always check for singularities in the integrand or limits of integration to determine if an integral is improper.