7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx

Verified step by step guidance

1. First, recognize that the integral is an improper integral because the lower limit is negative infinity: \(\int_{-\infty}^{-2} \frac{1}{x^{2}} \sin\left(\frac{\pi}{2}\right) \, dx\).
2. Note that \(\sin\left(\frac{\pi}{2}\right)\) is a constant value. Evaluate this constant to simplify the integral.
3. Rewrite the integral by factoring out the constant \(\sin\left(\frac{\pi}{2}\right)\), so the integral becomes \(\sin\left(\frac{\pi}{2}\right) \int_{-\infty}^{-2} \frac{1}{x^{2}} \, dx\).
4. Set up the improper integral as a limit: \(\lim_{t \to -\infty} \sin\left(\frac{\pi}{2}\right) \int_{t}^{-2} \frac{1}{x^{2}} \, dx\).
5. Evaluate the integral \(\int \frac{1}{x^{2}} \, dx\) by rewriting it as \(\int x^{-2} \, dx\) and using the power rule for integration, then apply the limits and take the limit as \(t\) approaches \(-\infty\) to determine if the integral converges or diverges.

Key Concepts

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, limits are used to approach the problematic points, determining if the integral converges to a finite value or diverges.

Behavior of the Integrand

Understanding the integrand's behavior, such as continuity and boundedness, is crucial. In this problem, the integrand includes 1/x² and a constant sine term, so analyzing how 1/x² behaves as x approaches negative infinity helps determine convergence.

Evaluation of Definite Integrals with Constants

When the integrand contains constant factors, they can be factored out of the integral to simplify evaluation. Recognizing that sin(π/2) is a constant (equal to 1) allows focusing on integrating 1/x² over the given interval.