What is the result of ∫ (1/x) dx?

The integral of ( \frac{1}{x} ) with respect to ( x ) is a fundamental result in calculus associated with the natural logarithm. When computing the integral, you have:

[

\int \frac{1}{x} , dx

]

The antiderivative of ( \frac{1}{x} ) is the natural logarithm function, specifically ( \ln|x| ). The absolute value is included to handle both the positive and negative values of ( x ), as the logarithm is only defined for positive arguments. The integral also includes a constant of integration, denoted by ( C ), which represents any constant value that can be added to the function since the derivative of a constant is zero.

Thus, the complete expression for the integral is:

[

\ln|x| + C

]

This result reflects the behavior of the logarithm function and adheres to the rules of integration for continuous functions. Recognizing that ( \ln(x) ) alone does not cover cases where ( x ) might be negative, it's essential to use the absolute value to ensure the expression remains valid across the domain of ( x ). Therefore, the answer is appropriately