About Integral Calculator
This integral calculator is free, needs no sign-up and runs in your browser, so the functions you type are never uploaded. Enter a function and it finds the indefinite integral with the constant of integration, or the exact value of a definite integral between two bounds.
It works through the problem the way a textbook would: substitution, integration by parts, partial fractions and the standard table are each named in the steps. For a definite integral it also shades the area on the graph, and bounds can be written as numbers, expressions like pi/2, or inf and -inf.
It is honest about what it cannot do. When no elementary antiderivative is found it says so instead of guessing, and when a value is a numeric estimate rather than an exact result it is labelled as one.
How to use Integral Calculator
- Type the function in the Function box, for example x*e^x, or use the key pad. The preview shows how it was read.
- Choose Indefinite for an antiderivative, or Definite to enter a lower and an upper bound.
- Write bounds as numbers or constant expressions such as pi/2. Type inf or -inf for an improper integral.
- Read the result card. A definite integral shows the exact value when one exists and a labelled numeric estimate otherwise.
- Check the graph: the function is drawn with its area shaded between the bounds, or with its antiderivative for an indefinite integral.
- Open 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 ideas behind the steps
An antiderivative F of f is a function with F' = f. The fundamental theorem of calculus says that the definite integral from a to b is F(b) - F(a).
When the integrand is a product, integration by parts moves the derivative from one factor to the other. When it contains a function inside another function, substitution undoes the chain rule.
integral x^n dx = x^(n+1)/(n+1) + C, n not -1
integral 1/x dx = ln|x| + C
integral u dv = u*v - integral v du
integral from a to b of f = F(b) - F(a)Worked examples
Integral of x*e^x gives x*e^x - e^x + C
Integration by parts with u = x and dv = e^x dx.
Integral of sin(x)^2 gives 1/2*x - 1/4*sin(2*x) + C
The power reduction identity turns sin^2 into a cosine of 2x.
Integral of x^2 from 0 to 3 gives 9
x^3/3 at 3 is 9 and at 0 is 0.
Integral of 1/x^2 from 1 to inf gives 1
An improper integral that settles on 1.
Integral of e^(-x^2) from 0 to 1 gives about 0.74682413
No elementary antiderivative exists, so this is a numeric estimate.
Good to know
What the calculator can and cannot do
It finds antiderivatives for functions built from powers, polynomials, rational functions that split into partial fractions, exponentials, logarithms and the trigonometric functions, including many products that need substitution or integration by parts.
Some integrals have no elementary antiderivative at all, such as e^(-x^2) or sin(x)/x. The calculator reports that no elementary rule was found. For those, enter bounds to get a numeric estimate, which is labelled as an estimate.
Integrals over an infinite range, or with a singular end such as 1/sqrt(x) from 0 to 1, are treated as improper integrals. Singular points inside the range are split out and taken as limits, so the integral of x^(-1/3) from -1 to 8 is 9/2. When no antiderivative exists the value comes from tanh-sinh (double-exponential) quadrature and is labelled as numeric. The calculator says "diverges" only when the integrand grows like 1/x or faster at a singular point, and an oscillating integral that does not settle is reported as not evaluated rather than as divergent.
Double integrals are not included. An iterated integral can be done by hand by integrating one variable at a time with the "With respect to" box.
About the constant of integration
Differentiating a constant gives zero, so an indefinite integral is a whole family of functions that differ by a constant. The result always ends with + C to show that. The constant cancels in a definite integral, which is why it does not appear there.
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.
How do I integrate a function?
Match the function to a rule: the power rule for x^n, a standard table entry for sin, cos and e^x, substitution when one function is inside another, integration by parts for products, and partial fractions for rational functions. The calculator names the rule it used.
What is the difference between an indefinite and a definite integral?
An indefinite integral is a family of functions, written with + C. A definite integral has two bounds and is a number: the signed area between the curve and the x axis.
Why does the calculator say no elementary rule was found?
Some functions, such as e^(-x^2), have no antiderivative made from elementary functions. The calculator then stops instead of guessing. A definite integral of the same function can still be estimated numerically.
How do I know if a result is exact or an estimate?
A numeric estimate is labelled "Numeric estimate" in the result card and is shown with the approximately sign. Exact results are shown in closed form.
Can I integrate to infinity?
Yes. Type inf or -inf as a bound. If the improper integral diverges, the calculator says so. If the integrand oscillates and the numeric value does not settle, it says it could not be evaluated rather than calling it divergent.
Does the calculator do double integrals?
No. For an iterated integral, integrate the inner variable first with "With respect to", then integrate that result in the other variable.
Why is there a + C in my answer?
Because functions that differ only by a constant have the same derivative, an indefinite integral stands for all of them. The C stands for that constant.
Popular Integral Calculator pages
- 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.
- 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.
- Integral of e^x*sin(x)Step-by-step integral of e^x*sin(x), with the method named and a graph of the function and its antiderivative.
- Integral of x^2*e^xStep-by-step integral of x^2*e^x, with the method named and a graph of the function and its antiderivative.
- Definite Integral CalculatorEnter a function and two bounds to get the exact area, with the region shaded on the graph.
- Antiderivative CalculatorEnter a function and get a function whose derivative it is, with the constant of integration.