Reciprocal Rule Integration Notes

Published on: September 29, 2026

This Reciprocal Rule Integration Calculator integrates one over x, and shows the working line by line. This is the single case the power rule cannot reach, because raising the exponent by one would give division by zero, so the answer is a natural logarithm instead. The absolute value keeps it valid for negative x as well. Following the same pattern every time keeps this exception clearly marked as one.

Step-by-step method

  1. Set up the problem.
  2. Write the reciprocal rule for integration.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Formula:

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the reciprocal rule for integration.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Identify the reciprocal function and any constant multiple.

In this problem: The function is \(\frac{1}{x}\).

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

Step 2

Step 2 - Write the reciprocal rule for integration.

In this problem: The integral of one over x is the natural logarithm of the absolute value of x. This is the special case the power rule cannot handle.

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the reciprocal with the logarithm and simplify.

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)

Final Answer

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)
See Example 2 Hide Example 2

Problem

\(\int \frac{5}{x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the reciprocal rule for integration.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Identify the reciprocal function and any constant multiple.

In this problem: The function is \(\frac{1}{x}\). The constant multiple \(5\) stays in front.

\(\int \frac{5}{x}\, dx\)

Step 2

Step 2 - Write the reciprocal rule for integration.

In this problem: The integral of one over x is the natural logarithm of the absolute value of x. This is the special case the power rule cannot handle.

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the reciprocal with the logarithm and simplify.

\(\int \frac{5}{x}\, dx = 5 \cdot \left(\ln\left|x\right|\right) = 5\ln\left|x\right| + C\)

Final Answer

\(\int \frac{5}{x}\, dx = 5\ln\left|x\right| + C\)
See Example 3 Hide Example 3

Problem

\(\int \frac{3}{x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the reciprocal rule for integration.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Identify the reciprocal function and any constant multiple.

In this problem: The function is \(\frac{1}{x}\). The constant multiple \(3\) stays in front.

\(\int \frac{3}{x}\, dx\)

Step 2

Step 2 - Write the reciprocal rule for integration.

In this problem: The integral of one over x is the natural logarithm of the absolute value of x. This is the special case the power rule cannot handle.

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the reciprocal with the logarithm and simplify.

\(\int \frac{3}{x}\, dx = 3 \cdot \left(\ln\left|x\right|\right) = 3\ln\left|x\right| + C\)

Final Answer

\(\int \frac{3}{x}\, dx = 3\ln\left|x\right| + C\)
See Example 4 Hide Example 4

Problem

\(\int - \frac{2}{x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the reciprocal rule for integration.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Identify the reciprocal function and any constant multiple.

In this problem: The function is \(\frac{1}{x}\). The constant multiple \(-2\) stays in front.

\(\int - \frac{2}{x}\, dx\)

Step 2

Step 2 - Write the reciprocal rule for integration.

In this problem: The integral of one over x is the natural logarithm of the absolute value of x. This is the special case the power rule cannot handle.

\(\int \frac{1}{x}\, dx = \ln\left|x\right| + C\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the reciprocal with the logarithm and simplify.

\(\int - \frac{2}{x}\, dx = -2 \cdot \left(\ln\left|x\right|\right) = -2\ln\left|x\right| + C\)

Final Answer

\(\int - \frac{2}{x}\, dx = -2\ln\left|x\right| + C\)