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

Definite Integral Calculator

Enter a function and two bounds to get the exact area, with the region shaded on the graph.

Kind of integral
x2+1x^{2}+1x2+1
A number, pi/2, inf or -inf.

Definite integral

∫03x2+1 dx=12\int_{0}^{3} x^{2}+1\,dx=12∫03​x2+1dx=12
  • f(x)
Table of values
Graph of x^2+1 with the area shaded: sampled values across the visible range
xf(x)
-0.751.5625
-0.31.09
0.151.0225
0.61.36
1.052.1025
1.53.25
1.954.8025
2.46.76
2.859.1225
3.311.89
3.7515.063

Step-by-step working

  1. 1Linearity
    ∫x2+1 dx\int x^{2}+1\,dx∫x2+1dx
    ↓
    ∫x2 dx+∫1 dx\int x^{2}\,dx+\int 1\,dx∫x2dx+∫1dx

    Integrate each term separately.

  2. 2Power rule
    ∫x2 dx\int x^{2}\,dx∫x2dx
    ↓
    x33\frac{x^{3}}{3}3x3​

    x^n integrates to x^(n+1) / (n + 1) for n other than -1.

  3. 3Constant rule
    ∫1 dx\int 1\,dx∫1dx
    ↓
    xxx
  4. 4Evaluate at the bounds
    [x33+x]03\left[\frac{x^{3}}{3}+x\right]_{0}^{3}[3x3​+x]03​
    ↓
    (12)−(0)\left(12\right)-\left(0\right)(12)−(0)

    Fundamental theorem of calculus: F(b) - F(a), with limits where the integrand is singular.

  5. 5Result
    ∫03x2+1 dx\int_{0}^{3} x^{2}+1\,dx∫03​x2+1dx
    ↓
    121212

Worked solution: integral of x^2 + 1 from 0 to 3

Worked steps

  1. 1Linearity
    ∫x2+1 dx\int x^{2}+1\,dx∫x2+1dx
    ↓
    ∫x2 dx+∫1 dx\int x^{2}\,dx+\int 1\,dx∫x2dx+∫1dx

    Integrate each term separately.

  2. 2Power rule
    ∫x2 dx\int x^{2}\,dx∫x2dx
    ↓
    x33\frac{x^{3}}{3}3x3​

    x^n integrates to x^(n+1) / (n + 1) for n other than -1.

  3. 3Constant rule
    ∫1 dx\int 1\,dx∫1dx
    ↓
    xxx
  4. 4Evaluate at the bounds
    [x33+x]03\left[\frac{x^{3}}{3}+x\right]_{0}^{3}[3x3​+x]03​
    ↓
    (12)−(0)\left(12\right)-\left(0\right)(12)−(0)

    Fundamental theorem of calculus: F(b) - F(a), with limits where the integrand is singular.

  5. 5Result
    ∫03x2+1 dx\int_{0}^{3} x^{2}+1\,dx∫03​x2+1dx
    ↓
    121212

Answer

∫03x2+1 dx=12\int_{0}^{3} x^{2}+1\,dx=12∫03​x2+1dx=12
  1. Tools
  2. Mathematics
  3. Integral Calculator
  4. Definite Integral Calculator

About Definite Integral Calculator

A definite integral gives the signed area between a curve and the x axis from one bound to another. This page opens with x^2 + 1 from 0 to 3, which equals 12. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The method is the fundamental theorem of calculus: find an antiderivative F, then compute F(upper) minus F(lower). For x^2 + 1 the antiderivative is x^3/3 + x, so the area is (9 + 3) - 0 = 12.

Bounds can be numbers, expressions such as pi/2, or inf and -inf for improper integrals. If no elementary antiderivative exists, the calculator falls back to a numeric estimate and labels it clearly.

How to use Definite Integral Calculator

  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.

Worked examples

  • Integral of x^2 + 1 from 0 to 3 → gives 12

    x^3/3 + x at 3 is 12, at 0 it is 0.

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 x^2 + 1 from 0 to 3?

The antiderivative is x^3/3 + x. At 3 it equals 12 and at 0 it equals 0, so the integral is 12.

Can a definite integral be negative?

Yes. Area below the x axis counts as negative, so the result is a signed area. The shaded region on the graph shows which parts add and which subtract.

Can I use infinity as a bound?

Yes. Type inf or -inf. The calculator checks that the improper integral settles on a finite value and says so if it does not.

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 1/xStep-by-step integral of 1/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.

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