95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2

Verified step by step guidance

1. Recognize that the integral is improper because the upper limit is infinity and the integrand involves a term \( \frac{\sin^2 x}{x^2} \), which is undefined at \( x = 0 \). So, we need to handle both the infinite limit and the behavior near zero carefully.
2. To approximate the integral numerically, first consider truncating the infinite upper limit to a large finite number \( M \), such that \( \int_0^\infty \frac{\sin^2 x}{x^2} dx \approx \int_0^M \frac{\sin^2 x}{x^2} dx \). Choose \( M \) large enough to capture the significant area under the curve.
3. Next, address the integrand's behavior near \( x = 0 \). Since \( \frac{\sin^2 x}{x^2} \) is indeterminate at zero, use the limit \( \lim_{x \to 0} \frac{\sin^2 x}{x^2} = 1 \) to define the value at zero or use a small positive number \( \epsilon \) as the lower limit instead of zero.
4. Apply a numerical integration method such as Simpson's rule, trapezoidal rule, or use a numerical integration function on a calculator or software to approximate \( \int_\epsilon^M \frac{\sin^2 x}{x^2} dx \). Make sure to use sufficiently small step sizes to improve accuracy.
5. Compare your numerical approximation to the exact value \( \frac{\pi}{2} \) to check the accuracy of your method. Adjust \( M \) and the step size if necessary to get a closer approximation.

Key Concepts

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with unbounded behavior. In this problem, the integral extends from 0 to infinity, requiring techniques to handle infinite limits or singularities to evaluate or approximate the integral.

Numerical Integration Methods

Numerical integration methods approximate definite integrals when exact solutions are difficult. Techniques like Simpson's rule, trapezoidal rule, or adaptive quadrature can be used, often with computational tools, to estimate the value of integrals, especially those with infinite limits.

Behavior of the Integrand and Convergence

Understanding the integrand's behavior, such as oscillations and decay, is crucial for convergence of improper integrals. Here, (sin²x)/x² oscillates but decays sufficiently fast, ensuring the integral converges to a finite value, which justifies numerical approximation methods.