7–64. Integration review Evaluate the following integrals.
47. ∫ dx / (x⁻¹ + 1)

Verified step by step guidance

1. Rewrite the integrand to simplify the expression. The denominator can be expressed as \( x^{-1} + 1 = \frac{1}{x} + 1 \). This simplifies the integral to \( \int \frac{dx}{\frac{1}{x} + 1} \).
2. Combine the terms in the denominator into a single fraction. The denominator becomes \( \frac{1 + x}{x} \), so the integral becomes \( \int \frac{dx}{\frac{1 + x}{x}} \).
3. Simplify the fraction by multiplying by the reciprocal of the denominator. This results in \( \int \frac{x}{1 + x} dx \).
4. Use substitution to simplify the integral. Let \( u = 1 + x \), so \( du = dx \). The integral becomes \( \int \frac{u - 1}{u} du \).
5. Split the fraction into two separate terms: \( \int \frac{u}{u} du - \int \frac{1}{u} du \). Simplify each term to \( \int 1 du - \int \frac{1}{u} du \), which can be integrated directly.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding the rules and techniques of integration, such as substitution and integration by parts, is essential for solving integral problems.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the context of integration, recognizing the form of a rational function is crucial for determining the appropriate method of integration. The integral in the question involves a rational function, which may require simplification or partial fraction decomposition to evaluate effectively.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. This method is particularly useful when dealing with complex functions or when the integrand can be transformed into a simpler form. By substituting a new variable, the integral can often be rewritten in a more manageable form, making it easier to evaluate.