The integral of 1/x is one of the most important formulas in calculus. Unlike most algebraic functions, it cannot be integrated using the standard power rule. Instead, it has a unique logarithmic result that every calculus student should know.
The formula is:
📘 Integral of 1/x Formula
The logarithmic rule used when the exponent is −1.
where:
- ln is the natural logarithm.
- |x| is the absolute value of x.
- C is the constant of integration.
Understanding why this formula is different—and how it compares with similar integrals like the integral of 1/x² and the integral of 1/(1+x²)—helps build a strong foundation in integration.
Integral of 1/x Formula
The standard formula is:
⚠ Why Doesn't the Power Rule Work?
The power rule says:
For 1/x, the exponent is −1.
Since dividing by zero is impossible, the power rule cannot be used. Instead:
✔ Quick Proof
The easiest way to verify the formula is by differentiation.Suppose:
Differentiate both sides:
Solved Examples
Similar Integrals You Should Know
Many students confuse the integral of 1/x with other common integrals. Here’s how they differ.
| Function | Integral | Rule Used |
|---|---|---|
| 1/x | ln|x| + C | Logarithmic Rule |
| 1/x² | −1/x + C | Power Rule |
| 1/(1+x²) | tan⁻¹(x) + C | Inverse Trigonometric Rule |
Integral of 1/x²
Rewrite it as:
Unlike the integral of 1/x, this one follows the normal power rule.
Integral of 1/(1+x²)
This is another standard integral.
This formula appears frequently in trigonometric substitution and differential equations.
Common Mistakes
❌ Using the Power Rule
1/x is the exception. Do not apply the standard power rule.
❌ Forgetting |x|
Always write ln|x|, not simply ln(x).
❌ Missing + C
Every indefinite integral must include the constant of integration.
📚 Integration Cheat Sheet
| Integral | Answer |
|---|---|
| ∫1/x dx | ln|x| + C |
| ∫1/x² dx | −1/x + C |
| ∫1/(1+x²) dx | tan⁻¹(x) + C |
| ∫k/x dx | k ln|x| + C |
Explore Related Math Guides:
Conclusion
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Frequently Asked Questions
Q1. What is the integral of 1/x?
Ans.
The integral of 1/x is:
Q2. Why is the integral of 1/x not found using the power rule?
Ans.
The power rule fails because the exponent is −1, which makes the denominator equal to zero. Since division by zero is undefined, the power rule cannot be applied. Instead, the integral is ln|x| + C.
Q3. What is the integral of 1/x²?
Ans.
The integral of 1/x² is:
Q4. What is the integral of 1/(1+x²)?
Ans.
The integral of 1/(1+x²) is:
Q5. Why do we write ln|x| instead of ln(x)?
Ans.
The absolute value ensures the logarithm is valid for both positive and negative values of x (except zero). Writing ln|x| makes the antiderivative correct over its entire domain.
