Simplify Your Workflow: Search MiniWebtool.
Add Extension
Home Page > Math > Calculus > Integral Calculator

Integral Calculator

Calculate definite and indefinite integrals with detailed step-by-step solutions, interactive function visualization, and comprehensive explanations for calculus students and professionals.

Integral Calculator

Try these examples:

x^2 + 3x + 1 sin(x) exp(x) x^2 from 0 to 1 sin(x) from 0 to pi 1/x
Integral Type:
Function f(x):
Variable:
💡
Syntax: Use ^ for powers, * for multiplication. Functions: sin, cos, tan, exp, ln, sqrt. Constants: pi, e.

Embed Integral Calculator Widget

About Integral Calculator

Welcome to the Integral Calculator, a powerful online tool for computing both definite and indefinite integrals with detailed step-by-step solutions. Whether you are a calculus student learning integration techniques, an engineer solving complex problems, or anyone needing to evaluate integrals quickly, this calculator provides accurate symbolic results with interactive visualizations to help you understand the integration process.

What is Integration?

Integration is one of the two fundamental operations of calculus (the other being differentiation). It represents the reverse process of differentiation and is used to find functions whose derivatives are known (antiderivatives) and to calculate areas, volumes, and accumulated quantities.

Indefinite Integral (Antiderivative)
$$\int f(x) \, dx = F(x) + C$$

Where $F(x)$ is the antiderivative of $f(x)$, meaning $F'(x) = f(x)$, and $C$ is the constant of integration representing the family of all antiderivatives.

The Definite Integral

The definite integral calculates the signed area between a function and the x-axis over a specific interval:

Definite Integral
$$\int_a^b f(x) \, dx = F(b) - F(a)$$

This formula, known as the Fundamental Theorem of Calculus, connects the concepts of antiderivatives and areas, allowing us to evaluate definite integrals using antiderivatives.

Common Integration Rules

Here are the fundamental integration formulas you need to know:

Power Rule
$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (n is not equal to -1)
Exponential
$\int e^x \, dx = e^x + C$
Natural Log
$\int \frac{1}{x} \, dx = \ln|x| + C$
Sine
$\int \sin(x) \, dx = -\cos(x) + C$
Cosine
$\int \cos(x) \, dx = \sin(x) + C$
Sum Rule
$\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$

How to Use This Calculator

  1. Choose integral type: Select whether you want to compute an indefinite integral (returns antiderivative + C) or a definite integral (returns a numerical value).
  2. Enter your function: Type the function using standard mathematical notation. Supported operations include polynomials (x^2), trigonometric functions (sin, cos, tan), exponential (exp, e^x), logarithmic (ln, log), and square root (sqrt).
  3. Specify the variable: Usually x, but you can use any single letter.
  4. For definite integrals: Enter the lower and upper limits. You can use numbers or expressions like pi, e, or sqrt(2).
  5. Calculate: View the result with step-by-step solution and interactive graphs.

Supported Function Syntax

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is one of the most important theorems in mathematics, establishing the connection between differentiation and integration.

Part 1: Derivative of an Integral

If $f$ is continuous on $[a, b]$ and $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$. This means the derivative of an integral recovers the original function.

Part 2: Evaluation of Definite Integrals

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then:

Fundamental Theorem (Part 2)
$$\int_a^b f(x) \, dx = F(b) - F(a)$$

This theorem allows us to evaluate definite integrals by finding an antiderivative and computing the difference at the bounds, rather than computing limits of Riemann sums.

Integration Techniques

Substitution (u-substitution)

For integrals of the form $\int f(g(x)) \cdot g'(x) \, dx$, let $u = g(x)$, then $du = g'(x) \, dx$. This transforms the integral into $\int f(u) \, du$, which may be easier to evaluate.

Integration by Parts

Based on the product rule for derivatives: $\int u \, dv = uv - \int v \, du$. This is useful for products of functions like $x \cdot e^x$ or $x \cdot \sin(x)$.

Partial Fractions

For rational functions (ratios of polynomials), decompose the fraction into simpler terms that can be integrated individually.

Trigonometric Substitution

For integrands containing $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$, use appropriate trigonometric substitutions.

Applications of Integration

Area Under a Curve

The most fundamental application: the definite integral $\int_a^b f(x) \, dx$ gives the signed area between the curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$.

Area Between Curves

The area between curves $y = f(x)$ and $y = g(x)$ from $a$ to $b$ is: $\int_a^b |f(x) - g(x)| \, dx$

Volumes of Revolution

Rotating a curve around an axis creates a solid whose volume can be calculated using the disk method: $V = \pi \int_a^b [f(x)]^2 \, dx$

Physics Applications

Frequently Asked Questions

What is an integral in calculus?

An integral is a fundamental concept in calculus that represents the accumulation of quantities, such as areas under curves or total change. The indefinite integral (antiderivative) finds a function whose derivative equals the original function. The definite integral calculates the signed area between a function and the x-axis over a specific interval. Integrals are the inverse operation of derivatives.

What is the difference between definite and indefinite integrals?

An indefinite integral finds the general antiderivative of a function and includes a constant of integration C. It is written as the integral of f(x) dx = F(x) + C. A definite integral evaluates the antiderivative at specific upper and lower bounds, giving a numerical value representing the signed area. The definite integral from a to b of f(x) dx equals F(b) minus F(a).

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus connects differentiation and integration. Part 1 states that if F(x) is the antiderivative of f(x), then the derivative of the integral from a to x of f(t)dt equals f(x). Part 2 states that the definite integral from a to b of f(x)dx equals F(b) minus F(a), where F is any antiderivative of f. This theorem allows us to evaluate definite integrals using antiderivatives.

What are the common integration techniques?

Common integration techniques include: Power Rule for polynomial terms, Substitution (u-substitution) for composite functions, Integration by Parts for products of functions, Partial Fractions for rational functions, Trigonometric Substitution for expressions with square roots of quadratics, and Trigonometric Identities for simplifying trigonometric integrands. The choice of technique depends on the form of the integrand.

What does the area under a curve represent?

The definite integral represents the signed area between a function and the x-axis. Areas above the x-axis are counted as positive, while areas below are counted as negative. This concept has many applications: in physics, the area under a velocity-time graph gives displacement; in economics, the area under a marginal cost curve gives total cost; in probability, the area under a probability density function gives probabilities.

Related Resources

Reference this content, page, or tool as:

"Integral Calculator" at https://MiniWebtool.com/integral-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 10, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

Related MiniWebtools:

Arc Length CalculatorFeatured
Convolution CalculatorFeatured
Curl CalculatorNew
Derivative Calculator
Divergence CalculatorNew
Double Integral Calculator
First-Order ODE SolverNew
Fourier Series Coefficients Calculator
Improper Integral CalculatorNew
Laplace Transform Calculator

Calculus:

Top & Updated:

Random PickerRandom Name PickerLine CounterBatting Average CalculatorRelative Standard Deviation CalculatorFPS ConverterSort NumbersERA CalculatorMAC Address GeneratorInstagram User ID LookupRemove SpacesWord to Phone Number ConverterMAC Address LookupFacebook User ID LookupJob FinderFeet and Inches to Cm ConverterSum CalculatorOPS CalculatorRandom Truth or Dare GeneratorPercent Off CalculatorRandom Quote GeneratorSHA256 Hash GeneratorSquare Root (√) CalculatorDoubling Time CalculatorLog Base 10 CalculatorBitwise CalculatorVertical Jump CalculatorNumber of Digits CalculatorRoman Numerals ConverterAudio SplitterMP3 LooperSlugging Percentage CalculatorSalary Conversion CalculatorSlope and Grade CalculatorOn Base Percentage CalculatorPhone Number ExtractorSaturn Return CalculatorRandom IMEI GeneratorRandom Poker Hand GeneratorAI Text HumanizerMerge VideosNumber to Word ConverterCaffeine Overdose CalculatorCompound Growth CalculatorSun, Moon & Rising Sign Calculator 🌞🌙✨Decimal to BCD ConverterImage ResizerRandom Birthday GeneratorFirst n Digits of PiWHIP CalculatorCompare Two StringsBinary to Gray Code ConverterCm to Feet and Inches ConverterGrade CalculatorBCD to Decimal ConverterAdd Prefix and Suffix to TextRandom Movie PickerOctal CalculatorRandom Fake Address GeneratorRandom Activity GeneratorWAR CalculatorOne Rep Max (1RM) CalculatorRandom Superpower GeneratorClothing Size ConverterFile Size ConverterVideo to Image ExtractorRandom Writing Prompt GeneratorInvisible Text GeneratorText FormatterYouTube Channel StatisticsLove Compatibility CalculatorCM to Inches ConverterRemove AccentOutlier CalculatorTime Duration CalculatorPercent Growth Rate CalculatorQuotient and Remainder CalculatorRandom Integer GeneratorStair CalculatorRandom Object GeneratorDay of Year CalendarList of Prime NumbersImage SplitterWord Ladder GeneratorAI Punctuation AdderRandom Number PickerRandom Loadout GeneratorRemove Leading Trailing SpacesGray Code to Binary ConverterCryptogram GeneratorExponential Decay CalculatorBingo Card GeneratorRandom Credit Card GeneratorArc Length CalculatorModulo CalculatorAdd Text to ImageDay of the Year Calculator - What Day of the Year Is It Today?Leap Years ListConnect the Dots GeneratorEmail ExtractorURL ExtractorAI ParaphraserSHA512 Hash GeneratorVideo CompressorBinary to BCD ConverterIP Address to Hex ConverterSort Lines AlphabeticallyHex to BCD ConverterBCD to Binary ConverterLottery Number GeneratorBCD to Hex ConverterMedian CalculatorStandard Error CalculatorList RandomizerBreak Line by CharactersAverage CalculatorPVIFA CalculatorReverse VideoHypotenuse CalculatorRemove Audio from VideoActual Cash Value CalculatorScientific Notation to Decimal ConverterNumber ExtractorAngel Number CalculatorLog Base 2 CalculatorRoot Mean Square CalculatorSum of Positive Integers CalculatorSHA3-256 Hash GeneratorAI Sentence Expander📅 Date CalculatorLbs to Kg ConverterHex to Decimal ConverterRandom Group GeneratorConvolution CalculatorMAC Address AnalyzerRandom String GeneratorAmortization CalculatorMarkup CalculatorPVIF CalculatorName Number CalculatorDecimal to Hex ConverterInstagram Font GeneratorSocial Media Image Size GuideTikTok Money CalculatorTwitter/X Character CounterTwitter/X Timestamp ConverterYouTube Watch Time CalculatorTwitch Earnings CalculatorYouTube Shorts Monetization CalculatorFacebook Ad Cost CalculatorSocial Media ROI CalculatorSocial Media Post Time OptimizerSocial Media Username CheckerCTR CalculatorROAS CalculatorInfluencer ROI CalculatorForce CalculatorAcceleration CalculatorVelocity CalculatorMomentum CalculatorProjectile Motion CalculatorKinetic Energy CalculatorPotential Energy CalculatorWork and Power CalculatorDensity CalculatorPressure CalculatorIdeal Gas Law CalculatorFree Fall CalculatorTorque CalculatorHorsepower CalculatorDilution CalculatorChemical Equation BalancerStoichiometry CalculatorPercent Yield CalculatorEmpirical Formula CalculatorBoiling Point CalculatorTitration CalculatorMole/Gram/Particle ConverterIrregular Polygon Area CalculatorFrustum CalculatorTorus Calculator3D Distance CalculatorGreat Circle Distance CalculatorCircumscribed Circle (Circumcircle) CalculatorInscribed Circle (Incircle) CalculatorAngle Bisector CalculatorTangent Line to Circle CalculatorHeron's Formula CalculatorCoordinate Geometry Distance CalculatorVolume of Revolution CalculatorSurface of Revolution CalculatorParametric Curve GrapherRiemann Sum CalculatorTrapezoidal Rule CalculatorSimpson's Rule CalculatorImproper Integral CalculatorL'Hôpital's Rule CalculatorMaclaurin Series CalculatorPower Series CalculatorSeries Convergence Test CalculatorInfinite Series Sum CalculatorAverage Rate of Change CalculatorInstantaneous Rate of Change CalculatorRelated Rates SolverOptimization Calculator (Calculus)Gradient Calculator (Multivariable)Divergence CalculatorCurl CalculatorLine Integral CalculatorSurface Integral CalculatorJacobian Matrix CalculatorNewton's Method CalculatorRREF Calculator (Row Echelon Form)Matrix Inverse CalculatorMatrix Multiplication CalculatorDot Product CalculatorCross Product CalculatorVector Magnitude CalculatorUnit Vector CalculatorAngle Between Vectors CalculatorNull Space CalculatorColumn Space CalculatorCramer's Rule CalculatorMatrix Diagonalization CalculatorQR Decomposition CalculatorCholesky Decomposition CalculatorMatrix Power CalculatorCharacteristic Polynomial CalculatorBayes' Theorem CalculatorF-Test / F-Distribution CalculatorHypergeometric Distribution CalculatorNegative Binomial Distribution CalculatorGeometric Distribution CalculatorExponential Distribution CalculatorWeibull Distribution CalculatorBeta Distribution CalculatorSpearman Rank Correlation CalculatorFisher's Exact Test CalculatorContingency Table CalculatorOdds Ratio CalculatorRelative Risk CalculatorEffect Size CalculatorPermutations with Repetition CalculatorModular Exponentiation CalculatorPrimitive Root CalculatorPerfect Number CheckerAmicable Number CheckerTwin Prime FinderMersenne Prime CheckerGoldbach Conjecture VerifierMöbius Function CalculatorEgyptian Fraction CalculatorFibonacci Number CheckerDigital Root CalculatorPartition Function CalculatorBoolean Algebra SimplifierKarnaugh Map (K-Map) SolverLogic Gate SimulatorGraph Coloring CalculatorTopological Sort CalculatorAdjacency Matrix CalculatorRecurrence Relation SolverInclusion-Exclusion CalculatorLinear Programming SolverTraveling Salesman Solver (TSP)Hamiltonian Path CheckerPlanar Graph CheckerNetwork Flow Calculator (Max Flow)Stable Marriage Problem SolverFirst-Order ODE SolverSecond-Order ODE SolverDirection Field / Slope Field PlotterEuler's Method CalculatorBernoulli ODE SolverSystem of ODEs SolverGroup Theory Order CalculatorRing and Field CalculatorJordan Normal Form CalculatorMatrix Exponential CalculatorTensor Product CalculatorFast Fourier Transform (FFT) CalculatorZ-Transform CalculatorNumerical Integration CalculatorTOML to JSON ConverterJSON to CSV ConverterXML to JSON ConverterSQL to MongoDB Query ConverterCSS Flexbox PlaygroundCSS Grid GeneratorJWT GeneratorBcrypt Hash Generator / CheckerColor Code Converter (All Formats)Git Command Generator.env File GeneratorLorem Picsum / Placeholder Image GeneratorText to Binary/Hex/ASCII ConverterSyllable CounterSentence CounterParagraph CounterSpeaking Time CalculatorReading Time CalculatorWhitespace VisualizerStrikethrough Text GeneratorTorque Converter (Nm, ft-lb, kgf-cm)Data Transfer Rate ConverterFuel Efficiency ConverterAstronomical Unit ConverterRing Size ConverterPaper Size ReferenceGas Mileage CalculatorEV Range CalculatorEV Charging Time Calculator0–60 / Quarter Mile CalculatorCar Lease CalculatorVehicle Towing Capacity CalculatorExposure Triangle CalculatorCrop Factor CalculatorMegapixel to Print Size CalculatorPhoto File Size EstimatorMusic BPM TapperMusic Key TransposerVideo Bitrate CalculatorSeed Germination Rate CalculatorFertilizer Calculator (NPK)Raised Bed Soil CalculatorFrost Date CalculatorLawn Fertilizer CalculatorCompost Calculator (C:N Ratio)Solar Panel CalculatorSolar ROI CalculatorHome Energy Audit CalculatorAppliance Energy Cost CalculatorWater Usage CalculatorElectricity Generation Cost CalculatorHeat Loss CalculatorFlight Distance CalculatorTravel Budget CalculatorJet Lag CalculatorPacking List GeneratorTip Splitter (Advanced)Lease vs Buy CalculatorHourly Rate Calculator (Freelancer)Invoice Late Fee CalculatorESPP CalculatorStock Split CalculatorOptions Probability CalculatorDollar to Gold ConverterBeam Load CalculatorPipe Flow CalculatorBolt Torque CalculatorSteel Weight CalculatorGravel, Sand & Topsoil CalculatorRandom Sentence GeneratorRandom Paragraph GeneratorRandom Math Problem GeneratorRandom Bible Verse GeneratorRandom Cat/Dog Name GeneratorRandom Debate Topic GeneratorBody Recomposition CalculatorAlcohol Calorie CalculatorMedication Dosage CalculatorPace to Calories CalculatorHydration CalculatorTrain Meeting Problem SolverAge Word Problem SolverMixture Problem SolverWork Rate Problem SolverDistance-Speed-Time Triangle CalculatorCoin Word Problem SolverNumber Bonds GeneratorCarry and Borrow VisualizerTimes Tables QuizMental Math Trainer