Calculus
Integral Calculator: Solve Definite And Indefinite Integrals With Steps
Last updated: October 2026
Written by Jessica Hill
An integral calculator finds the antiderivative of a function, or the area under its curve between two points. For example, ∫x² dx = x³/3 + C, and ∫ from 0 to 3 of x² dx = 9. It applies rules like the power rule, u-substitution and integration by parts, then shows each step.
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- Shows the working
- Runs on your device
Integration is the second big idea in calculus, after derivatives. This guide covers what integrals mean, the rules you use most, the four main methods, and how to apply integrals to real problems. Every result below was checked with the SymPy computer algebra system.
Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.
What An Integral Is
An integral has two related meanings. An indefinite integral is an antiderivative: a function whose derivative gives back the original function. A definite integral is a number that measures the signed area between a curve and the x-axis over an interval.
| Indefinite integral | Definite integral | |
|---|---|---|
| Written as | ∫f(x) dx | ∫ₐᵇ f(x) dx |
| Answer is | A function, plus C | A number |
| Example | ∫2x dx = x² + C | ∫₀³ 2x dx = 9 |
| Meaning | All functions whose derivative is 2x | Signed area under y = 2x from 0 to 3 |
The + C stands for any constant. Since the derivative of a constant is 0, x² + 5 and x² − 2 both have the derivative 2x. Writing + C covers all of them.
The ∫ sign is a stretched S for "sum." In ∫xy dx, y is treated as a constant.
How To Use The Integral Calculator
The integral calculator above handles indefinite, definite and double integrals.
- Pick a tab. Use Indefinite for an antiderivative, Definite for a number or area, and Double Integral for two variables.
- Type the function. Use ^ for powers, * for multiplication and parentheses for groups: x^2*sin(x), e^(2x), sqrt(1-x^2), 1/(x^2-1).
- Set the limits for a definite integral. You can type pi, e, inf or −inf.
Basic Integration Rules
Most integrals reduce to a short list of rules. Learn these and you can solve the majority of textbook problems.
| Function | Integral | Note |
|---|---|---|
| k (a constant) | kx + C | ∫5 dx = 5x + C |
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n + 1) + C | Power rule |
| 1/x | ln|x| + C | The n = −1 case |
| eˣ | eˣ + C | eˣ is its own integral |
| e^(kx) | e^(kx)/k + C | ∫e^(2x) dx = e^(2x)/2 + C |
| aˣ | aˣ/ln(a) + C | ∫2ˣ dx = 2ˣ/ln 2 + C |
| sin(x) | −cos(x) + C | Watch the minus sign |
| cos(x) | sin(x) + C | |
| sec²(x) | tan(x) + C | |
| 1/(1 + x²) | arctan(x) + C | |
| 1/√(1 − x²) | arcsin(x) + C |
Two more rules let you break problems apart. The sum rule says you can integrate term by term. The constant multiple rule says you can pull a number out front.
Example: ∫(x³ − 2x + 5) dx = x⁴/4 − x² + 5x + C. Each term uses the power rule, and only one + C is needed at the end.
The question "What is the integral of 3x³?" comes up often. Pull out the 3, apply the power rule to x³, and you get 3 × x⁴/4 = 3x⁴/4 + C.
How To Solve A Definite Integral
To solve a definite integral, find any antiderivative F(x), then subtract: ∫ₐᵇ f(x) dx = F(b) − F(a). This is the Fundamental Theorem of Calculus. The + C cancels when you subtract, so you can leave it out, as Paul's Online Math Notes at Lamar University explains.
Solve ∫₁² (3x² + 2x) dx:
- Antiderivative: F(x) = x³ + x².
- Upper limit: F(2) = 8 + 4 = 12.
- Lower limit: F(1) = 1 + 1 = 2.
- Subtract: 12 − 2 = 10.
Signed Area Vs Total Area
A definite integral counts area below the x-axis as negative. That means positive and negative parts can cancel.
For sin(x) from 0 to 2π, the hump above the axis has area 2 and the dip below has area 2. The integral is 2 + (−2) = 0, even though the curve clearly covers space. To get the total area, split the interval where the curve crosses the axis, take the absolute value of each piece, and add: 2 + 2 = 4.
If a problem says "find the area," it almost always wants total area. If it says "evaluate the integral," it wants the signed value. For flat shapes without curves, the area calculator gives the formula directly.
Integration Methods
When the basic rules do not fit, one of four methods usually does. Pick the method from the shape of the function.
U-Substitution
Use u-substitution when the function contains an inner piece and its derivative. It is the chain rule in reverse.
Solve ∫2x·cos(x²) dx:
- Let u = x². Then du = 2x dx.
- Replace: ∫cos(u) du.
- Integrate: sin(u) + C.
- Put x back: sin(x²) + C.
For a definite integral, either change the limits to u-values or switch back to x before plugging in. Mixing the two is a common error.
Integration By Parts
Use integration by parts for a product of two different kinds of functions, like x·eˣ or x·ln(x). The formula is ∫u dv = uv − ∫v du.
To pick u, many teachers use the LIATE order: Logarithmic, Inverse trig, Algebraic, Trig, Exponential. Choose u from whichever type comes first.
Solve ∫x·eˣ dx:
- u = x (algebraic), dv = eˣ dx.
- du = dx, v = eˣ.
- Apply the formula: x·eˣ − ∫eˣ dx.
- Finish: x·eˣ − eˣ + C = (x − 1)eˣ + C.
With ∫x·ln(x) dx, LIATE says u = ln(x), since logs come first. That gives (x²/2)ln(x) − x²/4 + C.
Partial Fractions
Use partial fractions for a fraction of polynomials whose bottom factors. Split it into simpler fractions, then integrate each one.
Solve ∫1/(x² − 1) dx:
- Factor: x² − 1 = (x − 1)(x + 1).
- Split: 1/(x² − 1) = ½·1/(x − 1) − ½·1/(x + 1).
- Integrate each: ½ln|x − 1| − ½ln|x + 1| + C.
Trig Identities And Trig Substitution
Powers of sine and cosine often need an identity first. For ∫sin²(x) dx, use sin²(x) = (1 − cos(2x))/2. The result is x/2 − sin(2x)/4 + C.
Roots like √(1 − x²) suggest a trig substitution, x = sin(θ), which gives ∫1/√(1 − x²) dx = arcsin(x) + C.
Improper Integrals
An improper integral has an infinite limit, or a function that blows up inside the interval. You solve it by replacing the problem spot with a variable and taking a limit.
| Integral | Work | Result |
|---|---|---|
| ∫₁^∞ 1/x² dx | lim (b → ∞) of [−1/x] from 1 to b = 0 − (−1) | Converges to 1 |
| ∫₁^∞ 1/x dx | lim (b → ∞) of ln(b) − ln(1) | Diverges (grows without limit) |
These two look almost the same, but one has a finite area and the other does not. In general, ∫₁^∞ 1/xᵖ dx converges only when p is greater than 1. Limits like these are covered step by step in the calculus solver.
Double Integrals
A double integral adds up a function of two variables over a region. Work from the inside out: integrate the inner variable while treating the other one as a constant, then integrate the result.
Solve ∫₀¹ ∫₀² (x + y) dy dx:
- Inner integral, in y: ∫₀² (x + y) dy = [xy + y²/2] from 0 to 2 = 2x + 2.
- Outer integral, in x: ∫₀¹ (2x + 2) dx = [x² + 2x] from 0 to 1 = 3.
- Result: 3.
With number limits, you can usually swap the order. When a limit depends on the other variable, the limits must change with the order.
When There Is No Exact Answer
Some functions have no antiderivative made of ordinary functions. The most famous is e^(−x²), the bell curve behind normal distributions. For these, calculators use numerical methods that approximate the area.
Simpson's rule is one of the most accurate simple methods. It fits small parabolas to the curve. With just four slices, it gives ∫₀¹ e^(−x²) dx ≈ 0.74686. The true value is about 0.74682, so the error is less than 0.0001.
A TI-84 does definite integrals this way. Press MATH and choose 9: fnInt, which Texas Instruments describes as a numerical integral over an interval. It gives a decimal, not a formula, so it cannot do indefinite integrals.
Real-World Uses Of Integrals
Integrals turn rates into totals and slices into whole shapes. Here are three common uses.
Area between two curves. Integrate the top curve minus the bottom one. Between y = x and y = x² from 0 to 1, the area is ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.
Distance from velocity. If a car's velocity is v(t) = 3t² + 2 meters per second, the distance in the first 4 seconds is ∫₀⁴ (3t² + 2) dt = 64 + 8 = 72 meters.
Volume of a solid. Spinning y = √x from 0 to 4 around the x-axis makes a bowl shape. Its volume is π∫₀⁴ x dx = 8π ≈ 25.13 cubic units.
How To Check An Integral
You can check any indefinite integral by taking the derivative of your answer. If you get the original function back, the integral is right.
Check ∫x·eˣ dx = (x − 1)eˣ + C. By the product rule, the derivative of (x − 1)eˣ is eˣ + (x − 1)eˣ = x·eˣ. It matches.
For a definite integral, sense-check the sign and size. x² on [0, 3] stays between 0 and 9, so the area must be between 0 and 27. An answer of 9 fits.
Common Integration Mistakes
Most integration errors come from small slips, not from picking the wrong method.
| Mistake | Example | Fix |
|---|---|---|
| Forgetting + C | ∫2x dx = x² | x² + C |
| Using the power rule on 1/x | ∫x⁻¹ dx = x⁰/0 | ∫1/x dx = ln|x| + C |
| Sign error with sine | ∫sin(x) dx = cos(x) + C | −cos(x) + C |
| Missing the 1/k with e^(kx) | ∫e^(2x) dx = e^(2x) + C | e^(2x)/2 + C |
| Subtracting in the wrong order | F(a) − F(b) | Always F(b) − F(a): upper minus lower |
| Old limits after u-substitution | Using x-limits with u | Change the limits to u-values, or switch back to x |
| Treating signed area as total area | ∫₀^(2π) sin(x) dx = 0, so "no area" | Split at zeros and add absolute values: 4 |
| Integrating a product term by term | ∫x·eˣ dx = (x²/2)·eˣ | Use integration by parts |
Summary
- An indefinite integral is an antiderivative plus C. A definite integral is a number equal to signed area.
- The power rule, 1/x → ln|x|, and the exponential and trig rules cover most basic problems.
- Use u-substitution for inner functions, by parts for products, partial fractions for factorable fractions, and identities for trig powers.
- Solve definite integrals with F(b) − F(a), and split at zeros when you need total area.
- Improper integrals use limits. Functions with no exact antiderivative use numerical methods like Simpson's rule.
- Check every indefinite integral by differentiating your answer.
Next step: type your integral into the integral calculator at the top of this page. For other math problems, try the math solver, or browse every tool on the Calc Solver homepage.
Frequently Asked Questions
What is the difference between an integral and a derivative?
They are opposite operations. A derivative measures how fast a function changes at each point, like speed from a position graph. An integral adds up small pieces to get a total, like distance from a speed graph. The Fundamental Theorem of Calculus links them: integrating a derivative gives back the original function, up to a constant.
What is the integral of 0?
The integral of 0 is a constant: ∫0 dx = C. The derivative of any constant is 0, so every constant function is an antiderivative of 0. As a definite integral, ∫ₐᵇ 0 dx is exactly 0, because a flat line on the x-axis encloses no area. This is why the + C in other answers can be any number.
Can every function be integrated?
Every continuous function has a definite integral over a closed interval, so the area always exists. But many functions have no antiderivative that can be written with ordinary functions like powers, logs, exponentials and trig. Examples include e^(−x²), sin(x)/x and √(1 + x³). For these, mathematicians use numerical methods or define new special functions, like the error function.
Why do we use dx in an integral?
The dx shows which variable you are integrating with respect to, and it stands for a very thin width. A definite integral adds up rectangles with height f(x) and width dx, then lets the width shrink toward zero. The dx also matters in u-substitution, where you replace dx with du divided by the derivative of the inner function.
Do I need an integral calculator for the AP Calculus exam?
The AP Calculus AB and BC exams have sections where a graphing calculator is allowed and sections where it is not. On calculator sections, a graphing calculator can find definite integrals numerically, but not symbolic antiderivatives. On non-calculator sections, you must integrate by hand. So an online tool is best for practice and checking, not for the test itself.
Try it with your own numbers
Use a free CalcSolver tool to check the examples in this guide.