Integral Calculator
Evaluate definite integrals numerically with adaptive Simpson quadrature (with exact antiderivatives for polynomials).
Details
Use ^ for powers, and sin, cos, ln or sqrt for functions.
The x value the area starts from. Sits under the ∫ sign.
The x value the area stops at. Sits above the ∫ sign.
Area under the curve
…
This integrates a function. Choose definite to get the area under the curve between two limits, or indefinite to get the antiderivative.
It returns the answer with the working shown, including the antiderivative it found first and how the two limits were substituted and subtracted.
What is an integral?
An integral adds up an infinite number of tiny pieces. Most often it is measuring the area underneath a curve.
Finding the area under a straight line is easy, because it makes triangles and rectangles. Under a curve there is no neat shape, so integration slices the region into strips so thin that the error disappears, then totals them.
It is also the exact reverse of differentiating, and that reverse has its own name: the antiderivative.
If differentiating x³ gives 3x², then the antiderivative of 3x² is x³. You are asking: *what function would I have to differentiate to end up with this one?* Finding an indefinite integral and finding an antiderivative are the same job, described from two directions.
The two ideas connect through one result. To get the area under a curve between two points, you find its antiderivative, put both points into it, and subtract. That link between areas and antiderivatives is the fundamental theorem of calculus, and it is why one tool answers both questions.
The two small numbers on the integral sign are the start and end points. The dx is not decoration: it says the strips are thin slices of x, and it tells you which letter is the variable.
What to enter
- Which integral do you want?
- Definite gives a number, the area between two points. Indefinite gives a function back, the general antiderivative.
- Your function
- Type it as you would say it: x^2, 3x, sin(x), e^x, 1/x. Use ^ for powers.
- From / to
- Only for a definite integral. The x value where the region starts and where it ends.
The two kinds, and how they differ
- Indefinite integral
- No limits on the sign. The answer is a function plus '+ C'. Asks: what would I have to differentiate to get this?
- Definite integral
- Has a start and end number. The answer is a single value, usually an area. No + C, because it cancels out.
- Why + C exists
- Differentiating destroys any constant, since the derivative of 5 is 0. So reversing the process cannot know what constant was there, and C stands in for every possibility.
What this assumes
The function is continuous across the range. A break or an infinite jump between the limits needs special handling.
Trigonometric functions use radians, as is standard in calculus.
How to calculate an integral
The power rule runs backwards. Where differentiating knocks the power down, integrating pushes it up.
- n + 1
- Add one to the power
- ÷ (n + 1)
- Then divide by that new power
- C
- The unknown constant, on indefinite integrals only
Raise the power by one. x² becomes x³.
Divide by the new power. So x² integrates to x³/3. Check it by differentiating back: the 3 comes down and cancels the ÷3, leaving x².
Add + C, if indefinite. Skip this only when you have limits.
For a definite integral, substitute and subtract. Put the top limit into your answer, then the bottom limit, and subtract the second from the first.
See a worked example: the area under x² from 0 to 3
- Function
- x²
- From
- 0
- To
- 3
Integrate first: raise the power to x³, then divide by 3, giving x³/3.
Put in the top limit: 3³ ÷ 3 = 27 ÷ 3 = 9.
Put in the bottom limit: 0³ ÷ 3 = 0.
Subtract: 9 − 0 = 9.
No + C here, because it would appear in both substitutions and cancel itself out.
The area is 9
Frequently asked questions
A definite integral has numbers on the integral sign and gives you a single value, normally an area. An indefinite one has no limits and gives you a function back.
If the question says 'find the area between x = 1 and x = 4', that is definite. If it just says 'integrate', it is indefinite and needs + C.
Because differentiating erases constants. x² + 3, x² − 40 and plain x² all differentiate to 2x, so going backwards from 2x cannot tell which one you started with.
C covers all of them at once. Leave it off an indefinite integral and you will lose marks, every time.
Yes. Any part of the curve below the x-axis counts as negative area, so a definite integral can come out negative or even zero.
If you want physical area rather than signed area, split the calculation where the curve crosses the axis and make each piece positive before adding.
It marks which variable you are slicing along, and closes the expression the way a full stop closes a sentence.
It matters when more than one letter is present: integrating 3ax dx treats a as a fixed number, while da would treat x as fixed instead.
It is the function you would have to differentiate to get the one you started with. The antiderivative of 2x is x², because differentiating x² gives 2x.
'Antiderivative' and 'indefinite integral' mean the same thing. The first name describes what it does, the second describes how it is written.
Strictly there is not one antiderivative but a whole family of them, all differing by a constant, which is exactly what the + C stands for.
Differentiate it. Integration and differentiation undo each other, so differentiating your answer should give back exactly what you started with.
This works for indefinite integrals in seconds and catches most slips, since the + C simply disappears again.
Problems people actually run into
Dropping the + C
The most common lost mark in the whole topic. An indefinite integral without + C is an incomplete answer, not a slightly untidy one.
The habit worth building: write + C as soon as you finish integrating, before you tidy anything up, so it is never an afterthought.
Subtracting the limits the wrong way round
It is always top limit minus bottom limit. Doing it in reverse flips the sign of the whole answer.
An area that comes out negative when the curve is clearly above the axis is nearly always this mistake rather than a genuine negative region.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: September 3, 2026