The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]

Verified step by step guidance

1. Identify the integral to solve: \(\int_{2}^{4} \frac{dt}{t \sqrt{t^{2} - 4}}\).
2. Recognize that the integrand contains a square root of the form \(\sqrt{t^{2} - a^{2}}\), suggesting a trigonometric substitution. Use the substitution \(t = 2 \sec(\theta)\), where \(a = 2\).
3. Compute the differential \(dt\) in terms of \(d\theta\): since \(t = 2 \sec(\theta)\), then \(dt = 2 \sec(\theta) \tan(\theta) d\theta\).
4. Rewrite the integral in terms of \(\theta\) by substituting \(t\), \(dt\), and simplifying the expression under the square root: \(\sqrt{t^{2} - 4} = \sqrt{4 \sec^{2}(\theta) - 4} = 2 \tan(\theta)\).
5. Change the limits of integration from \(t\) to \(\theta\) using \(t = 2 \sec(\theta)\), then simplify the integral and integrate with respect to \(\theta\).

Key Concepts

Improper Integrals and Convergence

Understanding when an integral converges is crucial, especially for integrals with infinite limits or integrands with singularities. In this problem, the integral is definite and the integrand is continuous on [2,4], ensuring convergence. Recognizing convergence allows safe evaluation without concern for divergence.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(t² − a²). By substituting t = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate. This method transforms the integral into a trigonometric integral that can be integrated directly.

Integration Techniques for Rational Functions

Integrals involving rational functions combined with radicals often require algebraic manipulation or substitution to simplify. Recognizing the structure of the integrand, such as t in the denominator and a radical in the denominator, guides the choice of substitution and integration steps to find an explicit antiderivative.