Skip to content
ToolUlkaa Tool
All tools

Mathematics

Percentage CalculatorFraction CalculatorEquation SolverTensorMatrix CalculatorUnit ConverterNumber Base ConverterDerivative CalculatorIntegral CalculatorLimit CalculatorAlgebra CalculatorQuadratic Formula CalculatorFactoring CalculatorLogarithm CalculatorSquare Root CalculatorLong Division CalculatorSignificant Figures CalculatorLCM and GCF CalculatorPrime Factorization CalculatorTriangle CalculatorPythagorean Theorem CalculatorSlope CalculatorArea CalculatorVolume CalculatorScientific CalculatorGraphing Calculator

Categories

AcademicTextMathematicsStatisticsMachine LearningAIData ScienceFile
All tools
Mathematics

Integral of 1/x

Step-by-step integral of 1/x, with the method named and a graph of the function and its antiderivative.

Kind of integral
1x\frac{1}{x}x1​

Antiderivative

∫1x dx=ln⁡(∣x∣)+C\int \frac{1}{x}\,dx=\ln\left(\left|x\right|\right)+C∫x1​dx=ln(∣x∣)+C
  • f(x)
  • F(x), with C = 0
Table of values
Graph of 1/x: sampled values across the visible range
xf(x)F(x), with C = 0
-6-0.166671.7918
-4.8-0.208331.5686
-3.6-0.277781.2809
-2.4-0.416670.87547
-1.2-0.833330.18232
0——
1.20.833330.18232
2.40.416670.87547
3.60.277781.2809
4.80.208331.5686
60.166671.7918

Step-by-step working

  1. 1Integral of 1/x
    ∫1x dx\int \frac{1}{x}\,dx∫x1​dx
    ↓
    ln⁡(∣x∣)\ln\left(\left|x\right|\right)ln(∣x∣)
  2. 2Add the constant
    ln⁡(∣x∣)\ln\left(\left|x\right|\right)ln(∣x∣)
    ↓
    ln⁡(∣x∣)+C\ln\left(\left|x\right|\right)+Cln(∣x∣)+C

    Every antiderivative differs by a constant.

Worked solution: integral of 1/x

Worked steps

  1. 1Integral of 1/x
    ∫1x dx\int \frac{1}{x}\,dx∫x1​dx
    ↓
    ln⁡(∣x∣)\ln\left(\left|x\right|\right)ln(∣x∣)
  2. 2Add the constant
    ln⁡(∣x∣)\ln\left(\left|x\right|\right)ln(∣x∣)
    ↓
    ln⁡(∣x∣)+C\ln\left(\left|x\right|\right)+Cln(∣x∣)+C

    Every antiderivative differs by a constant.

Answer

∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx=\ln\left|x\right|+C∫x1​dx=ln∣x∣+C
  1. Tools
  2. Mathematics
  3. Integral Calculator
  4. Integral of 1/x

About Integral of 1/x

The integral of 1/x with respect to x is ln(abs(x)) + C. The power rule fails for x^(-1) because it would divide by zero. Instead, 1/x is the derivative of ln|x|, so this is a standard integral. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The calculator below opens with this problem already worked, and the full steps are written out above. Switch to Definite and give two bounds to see the area, shaded on the graph. For example, from 1 to e the area is 1.

You can always check the answer by differentiating it: the derivative of ln(abs(x)) is 1/x.

How to use Integral of 1/x

  1. 1Type the function in the Function box, for example x*e^x, or use the key pad. The preview shows how it was read.
  2. 2Choose Indefinite for an antiderivative, or Definite to enter a lower and an upper bound.
  3. 3Write bounds as numbers or constant expressions such as pi/2. Type inf or -inf for an improper integral.
  4. 4Read the result card. A definite integral shows the exact value when one exists and a labelled numeric estimate otherwise.
  5. 5Check the graph: the function is drawn with its area shaded between the bounds, or with its antiderivative for an indefinite integral.
  6. 6Open the steps to see which method was used, and copy the answer with the copy button.

Features

  • Indefinite integrals with the constant of integration.
  • Definite integrals with exact values, including bounds written with pi and with inf or -inf.
  • Step-by-step working that names the rule: power rule, substitution, integration by parts, partial fractions and trigonometric identities.
  • A shaded area under the curve for definite integrals, on a graph you can zoom and drag.
  • A clear label when the result is a numeric estimate instead of an exact value.
  • An honest "no elementary rule found" message instead of a guess.
  • A live preview and a math key pad, so input is easy on a phone.
  • Checks improper integrals and says when one does not settle on a finite value.

The result and the method behind it

The power rule fails for x^(-1) because it would divide by zero. Instead, 1/x is the derivative of ln|x|, so this is a standard integral.

integral 1/x dx = ln(abs(x)) + C

Worked examples

  • Integral of 1/x dx → gives ln(abs(x)) + C

    The power rule fails for x^(-1) because it would divide by zero. Instead, 1/x is the derivative of ln|x|, so this is a standard integral.

  • Integral of 1/x from 1 to e → gives 1

    The antiderivative at the upper bound minus the antiderivative at the lower bound.

Privacy: how it works

  • The integral is worked out by code running in your browser tab. Your functions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Differentiate your answer with the derivative calculator to check it. You should get the original function back.
  • For an area between a curve and the x axis, remember that parts below the axis count as negative in a definite integral.
  • An improper integral with a bound at infinity only has a value if the function dies away fast enough. 1/x^2 does, 1/x does not.
  • Use * between factors when you can, for example x*e^x, and write powers of e as e^x.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your functions are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your functions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What is the integral of 1/x?

It is ln(abs(x)) + C. The power rule fails for x^(-1) because it would divide by zero. Instead, 1/x is the derivative of ln|x|, so this is a standard integral.

What is the definite integral of 1/x from 1 to e?

Evaluate the antiderivative ln(abs(x)) at the two bounds and subtract. The result is 1.

Why is there an absolute value in ln|x|?

The function 1/x is defined for negative x too, and the derivative of ln(-x) is also 1/x. The absolute value makes the antiderivative valid on both sides of 0.

More Integral Calculator pages

  • Integral CalculatorFind an antiderivative or the area between two bounds, with the method named at every step.
  • Integral of ln(x)Step-by-step integral of ln(x), with the method named and a graph of the function and its antiderivative.
  • Integral of sin^2(x)Step-by-step integral of sin^2(x), with the method named and a graph of the function and its antiderivative.
  • Integral of x*e^xStep-by-step integral of x*e^x, with the method named and a graph of the function and its antiderivative.
  • Integral of tan(x)Step-by-step integral of tan(x), with the method named and a graph of the function and its antiderivative.
  • Integral of sec(x)Step-by-step integral of sec(x), with the method named and a graph of the function and its antiderivative.
  • Integral of e^(-x^2)Why e^(-x^2) has no closed-form antiderivative, and its numeric value over any range.
  • Integral of x*sin(x)Step-by-step integral of x*sin(x), with the method named and a graph of the function and its antiderivative.
  • Integral of 1/(1+x^2)Step-by-step integral of 1/(1+x^2), with the method named and a graph of the function and its antiderivative.
  • Integral of sqrt(x)Step-by-step integral of sqrt(x), with the method named and a graph of the function and its antiderivative.
  • Integral of cos^2(x)Step-by-step integral of cos^2(x), with the method named and a graph of the function and its antiderivative.
  • Integral of 1/(x^2-1)Step-by-step integral of 1/(x^2-1), with the method named and a graph of the function and its antiderivative.
  • Integral of x/(x^2+1)Step-by-step integral of x/(x^2+1), with the method named and a graph of the function and its antiderivative.

Ulkaa Apps

  • Ulkaa
  • Calendar
  • Task
  • Note
  • Disk
  • Call
  • Contact
  • Mail
  • Finance
  • Health
  • Office
  • Work
  • Workspace
  • Halum
  • Jaal
  • Islam
  • Game
  • Weather
  • Campus
  • News
  • Developer
  • Help
  • Account
About·Privacy·Terms·© 2026 Ulkaa

What is Ulkaa Tool?

Ulkaa Tool is a free collection of fast, privacy-friendly online tools from Ulkaa Network. Calculate your GPA and CGPA, edit rich text, count words and characters, replace text and more — all in your browser.

Every tool runs entirely in your browser — no sign-up, and your data never leaves your device.

Academic

  • GPA Calculator
  • CGPA Calculator
  • Citation Generator
  • Quiz Maker

Text

  • Rich Text Editor
  • Word Counter
  • Character Counter
  • Text Replacement

Ulkaa

  • Ulkaa
  • Ulkaa News
  • Ulkaa Campus
  • Ulkaa Tool
  • Ulkaa Game
  • Ulkaa Weather
  • Ulkaa Islam

Company

  • About Ulkaa
  • Careers
  • Support
  • Help
  • Privacy Policy
  • Terms of Service
  • Disclaimer
  • Sitemap
© 2015-26, Ulkaa. All rights reserved.
Facebook PageLinkedIn CompanyX (Twitter) ProfileInstagram ProfileGitHub Account