Ohio Assessments for Educators (OAE) Mathematics Practice Exam 2025 - Free Practice Questions and Study Guide

The integral of \( \frac{1}{x} \, dx \) is derived from the fundamental properties of logarithmic functions. Specifically, when calculating the indefinite integral of \( \frac{1}{x} \), it follows the rule that the integral of \( \frac{1}{u} \, du \) results in \( \ln|u| + C \), where \( C \) represents the constant of integration.

In this case, since \( u \) is replaced by \( x \), the integral evaluates to \( \ln|x| + C \). The use of the absolute value is crucial because the natural logarithm is defined only for positive arguments, ensuring the function remains valid even when \( x \) is negative.

This understanding links directly to the properties of logarithmic functions, reinforcing that the integral captures the behavior of the \( \frac{1}{x} \) function as it approaches zero and extends in both positive and negative directions along the x-axis. Hence, the integral's result, \( \ln|x| \), is a natural representation of the accumulated area under the curve of \( \frac{1}{x} \).