Reciprocal Rule Integration Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Reciprocal Rule Integration Calculator helps you integrate reciprocal functions such as 1/x or 5/x. The power rule cannot handle the exponent -1, so this special case integrates to the natural logarithm of the absolute value of x.

Step-by-step method

  1. Identify the reciprocal function and any constant multiple.
  2. Write the reciprocal rule for integration.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Formula:

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

Example 1:

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

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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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:

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