Integral of 1/x
Antiderivative of the reciprocal function.
Formula
Variables
Example
Gives the natural log
How it works
The function 1/x is special because its antiderivative is the natural logarithm, so integrating a reciprocal turns growth measured by multiplication into addition. The absolute value is needed because ln(x) is only defined for positive x, while 1/x is defined for both positive and negative x except at 0; using |x| makes the rule valid on either side of zero. The constant C appears because many different antiderivatives have the same derivative.
When to use it
Use this when you need the antiderivative of 1/x or of expressions that can be rewritten as a constant multiple of 1/x. It also appears in logarithmic integration, separable differential equations, and anywhere rates proportional to 1/x are being accumulated.
Worked example, step by step
Find \int \frac{1}{7x}\,dx Step 1: Factor out the constant 1/7: \int \frac{1}{7x}\,dx = \frac{1}{7}\int \frac{1}{x}\,dx Step 2: Apply the rule \int \frac{1}{x}\,dx = \ln|x| + C: \frac{1}{7}\int \frac{1}{x}\,dx = \frac{1}{7}(\ln|x| + C) Step 3: Simplify the constant term into a new constant: \frac{1}{7}\ln|x| + C Final answer: \frac{1}{7}\ln|x| + C
Common mistakes to avoid
- Writing ln(x) instead of ln|x|, which fails for negative x
- Forgetting the constant of integration C
- Treating 1/x like x^-1 and expecting a power rule result such as x^0/0
Did You Know?
The integral of 1/x is ln|x|, the elegant patch for the one gap left by the power rule.
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