76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
82. ∫ [dx / (x√(1 + 2x))]

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{dx}{x \sqrt{1 + 2x}}\).
2. Recognize that the expression under the square root, \$1 + 2x\(, suggests a substitution to simplify the radical. Let \)t = \sqrt{1 + 2x}\(, so that \)t^2 = 1 + 2x$.
3. Differentiate both sides of \(t^2 = 1 + 2x\) with respect to \(x\) to find \(dx\) in terms of \(dt\): \$2t \frac{dt}{dx} = 2\(, which implies \)\frac{dt}{dx} = \frac{1}{t}\(, so \)dx = t \, dt$.
4. Express \(x\) in terms of \(t\) from \(t^2 = 1 + 2x\): \(x = \frac{t^2 - 1}{2}\). Substitute \(x\) and \(dx\) back into the integral to rewrite it entirely in terms of \(t\).
5. Simplify the integral after substitution and then proceed to use partial fractions or other integration techniques as appropriate to evaluate the integral.

Key Concepts

Substitution Method (Change of Variables)

This technique involves replacing a complicated expression with a simpler variable to make integration easier. For example, setting u = 1 + 2x transforms the integral into a function of u, simplifying the square root and rational expressions. It is often the first step before applying other integration methods.

Partial Fraction Decomposition

Partial fractions break down a complex rational function into simpler fractions that are easier to integrate. After substitution, the integrand often becomes a rational function suitable for this method, allowing the integral to be expressed as a sum of simpler terms.

Integration of Rational Functions Involving Roots

Integrals containing roots like √(1 + 2x) often require manipulation to rewrite the integrand in a rational form. Understanding how to handle expressions with roots and convert them into integrable forms is essential, often combining substitution and partial fractions.