Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x² - 4) / x dx

Verified step by step guidance

1. Recognize that the integral is of the form \(\int \frac{\sqrt{x^2 - a^2}}{x} \, dx\) where \(a = 2\). This suggests using a trigonometric substitution to simplify the square root expression.
2. Use the substitution \(x = 2 \sec \theta\), which implies \(dx = 2 \sec \theta \tan \theta \, d\theta\). This substitution transforms \(\sqrt{x^2 - 4}\) into \(\sqrt{4 \sec^2 \theta - 4} = 2 \tan \theta\).
3. Rewrite the integral in terms of \(\theta\) by substituting \(x\), \(dx\), and \(\sqrt{x^2 - 4}\) with their expressions involving \(\theta\). The integral becomes \(\int \frac{2 \tan \theta}{2 \sec \theta} \cdot 2 \sec \theta \tan \theta \, d\theta\).
4. Simplify the integrand by canceling terms and combining like factors. This will reduce the integral to a function involving trigonometric expressions in \(\theta\) that are easier to integrate.
5. Integrate the simplified trigonometric expression with respect to \(\theta\), then substitute back \(\theta = \sec^{-1}(x/2)\) to express the answer in terms of \(x\).

Key Concepts

Integration Using Standard Integral Formulas

Many integrals can be evaluated by recognizing their form and applying standard integral formulas from tables. These tables list integrals of common functions, including those involving roots and rational expressions, which simplifies the integration process without performing substitution or integration by parts manually.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(x² - a²). By substituting x = a sec(θ), the radical simplifies to a trigonometric function, making the integral easier to evaluate. This method is often referenced in integral tables for such forms.

Algebraic Manipulation of the Integrand

Before applying integral formulas, it is important to manipulate the integrand algebraically to match a standard form. This may involve factoring, simplifying fractions, or rewriting expressions to fit the integral forms listed in the table, ensuring the correct formula is applied.