Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the integral of (1/x)dx?

ln|x|
x
1/x
e^x

The correct answer is: ln|x|

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} $.