Integral Calculator
Type an integrand and the antiderivative appears, with the expression the page actually read printed underneath it so a missing bracket or an implied multiplication can never pass for the question you asked. Leave the bounds empty for an antiderivative, fill them in for a definite integral, or switch to a double integral where the inner bounds may lean on the outer variable. The symbolic work runs in a background worker with a two second stop on it, which is why an integrand that would take a long search stops the worker rather than the page. Where there is no elementary antiderivative the page says so and names the special function that would be needed, instead of printing an approximation as though it were the answer, and the Riemann and trapezoid methods sit alongside for a number over an interval, drawn as rectangles or strips with the subinterval count on screen.
- Result
- Variable
- Bounds
- Numeric
What you can type
Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x, 3 sin(x) and 2(x + 1) all read the way you would write them on paper, and the stage prints back what that came to. Leave both bounds empty for an antiderivative, or fill them in for a definite integral. Powers go up to 64, and the expression can be up to 500 characters long.
Common questions
- What does the + C mean in the answer?
- It is the constant of integration. Every function that differs by a constant has the same derivative, so an indefinite integral is a family of curves rather than one, and x^3/3, x^3/3 + 1 and x^3/3 - 40 all differentiate back to x^2. The page prints the + C on indefinite results and leaves it off definite ones, where the constant cancels between the two bounds.
- Why does it say there is no elementary antiderivative?
- Because for some integrands there is no answer in the ordinary functions, and exp(x^2) is the standard example. Writing its antiderivative down needs the error function erf, which is defined by the integral itself, so it names no method you could carry out by hand. The page says that plainly and names the function rather than printing a number in its place. Set two bounds and the Trapezoid method will give you a value over that interval instead.
- Does it show the steps?
- No. It gives the antiderivative, the definite value, the decimal and the expression it read, and it does not claim a derivation: the library behind it searches for a form that differentiates back, which is not the substitution or parts argument a tutor would write out. If you need the working, the result here is what to check your own working against.
- How do I write the expression?
- Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs and log. ln is read as log, the natural logarithm, and arcsin, arccos and arctan are read as asin, acos and atan. You can leave multiplication out where it is obvious, so 2x and 3 sin(x) both work, and the stage shows what that came to.
- Can it do double integrals?
- Yes. Switch Integrals to Double and you get a second variable with its own pair of bounds. The inner integral runs first, so its bounds may use the outer variable, which is how a region rather than a rectangle is described; the outer bounds may only use numbers and constants, and the page refuses them if they name the inner variable. With no bounds at all you get the iterated antiderivative, for example x^2 y^2 over 4 for x times y.
- What is the difference between the Riemann and Trapezoid methods?
- Both are numerical: they divide the interval into equal subintervals and add up areas, from 1 to 10,000 of them. Riemann uses rectangles whose height is the integrand at the left edge, the right edge or the middle of each one; Trapezoid joins the two edge heights with a straight line, which is the same as counting every interior sample once and the two end samples half each. Neither is the exact integral, and the page labels them approximations: x^2 from 0 to 1 with two strips comes to 0.375, where the exact value is one third.
- Why does tan(x) from 0 to pi show no value?
- Because the library's figure and the sampling disagree. After a definite integral comes back, the page samples the same integrand at 200 and 400 steps across the interval; when those two runs do not settle on the same number, or when a sample strictly between the bounds has no value at all, there is a pole in the way that a closed form taken at the two ends has stepped over. tan(x) from 0 to pi is the clear case: it has a pole at pi/2 and does not converge, so the page shows no figure in the result or the numeric cell, the copied line says the same rather than giving a value, and it tells you to split the interval at the pole. A bound the sampling cannot use, as at 0 in 1/sqrt(x) from 0 to 1, is a different thing: that integral does converge, so the value stands and the page says only that the check could not be taken.
- Does my expression leave the browser?
- No. There is no upload and no account: the reading, the sampling and the symbolic work all happen in the tab, the last one in a worker built from this page's own code with the algebra library loaded from this site. Your integrand, variable and bounds are kept in your browser's local storage so the page opens where you left it, and the Start over button at the top of the page forgets them.
Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.