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.

Is a definite integral the area under the curve?

It is the signed area: the part of the curve above the axis counts as positive and the part below as negative, which is why the drawing shades both. So sin(x) from 0 to 2*pi comes to 0, the two halves cancelling. For the total area regardless of sign, integrate the absolute value instead: abs(sin(x)) over the same bounds gives 4.

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.

Does it handle integrals that need substitution or integration by parts?

It finds the antiderivative whatever method a person would pick by hand: x*exp(x), a by parts integral, comes back as -e^x + e^x*x + C, and x*cos(x^2), a substitution, as sin(x²)/2 + C. It does not say which substitution or which parts it used, and a rational function that needs partial fractions can come back unfinished, in which case the page says no antiderivative was found.

How do I write the expression?

Use the supported functions listed under What you can type. Multiplication can be left out where it is clear: 2x and 3 sin(x) both work, and the stage shows how the expression was read. A letter other than the integration variable is held as a constant, so x*y with respect to x gives x²*y/2 + C.

Can it do double and triple integrals?

Yes. Switch Integrals to Double or Triple and each extra variable gets its own pair of bounds. The inner integral runs first, so its bounds may use the variables outside it, which is how a region rather than a box is described; the outer bounds take numbers and constants only, and the page refuses a bound that names a variable already integrated. With no bounds at all you get the iterated antiderivative, for example x²*y²/4 for x times y. A double or triple value is also sampled over its region: a figure the sampling contradicts is not shown, and a numerical value is cut to the digits the sampling confirms.

Can it work in polar, cylindrical or spherical coordinates?

Yes, with the extra factor typed into the integrand, because the page integrates exactly what is in the box and adds nothing. Use single letters, such as r for the radius, t for the angle θ and p for the angle φ, then multiply by r for polar or cylindrical coordinates and by r^2*sin(p) for spherical ones. With the integrand r, r from 0 to 1 and t from 0 to 2*pi, a double integral gives pi, the area of the unit disc.

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.

How do I do an integral on a TI-84?

Press MATH and choose 9:fnInt(, then give it the expression, the variable and the two bounds, as in fnInt(X^2,X,0,3), which returns 9. The TI-84 works the value out numerically and has no antiderivative function, so an indefinite integral is the part it cannot do. This page gives both, in any current browser on a phone, tablet or computer, with nothing to install.

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.

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 and see the area under the curve shaded, or switch to a double or triple integral, where an inner bound may lean on the variables outside it. 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.

IntegralWorking∫ x² dx · read as x^2 · symbolic

Integrand

Method

Integrals

Result: Working
Variable
Bounds
Numeric: n/a

x^2 dxindefinite
x³/3 + C
x^2 from 0 to 3definite
9
exp(x^2) dxno elementary form
named, not approximated

Accuracy. 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.

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.
Is a definite integral the area under the curve?: It is the signed area: the part of the curve above the axis counts as positive and the part below as negative, which is why the drawing shades both. So sin(x) from 0 to 2*pi comes to 0, the two halves cancelling. For the total area regardless of sign, integrate the absolute value instead: abs(sin(x)) over the same bounds gives 4.
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.
Does it handle integrals that need substitution or integration by parts?: It finds the antiderivative whatever method a person would pick by hand: x*exp(x), a by parts integral, comes back as -e^x + e^x*x + C, and x*cos(x^2), a substitution, as sin(x²)/2 + C. It does not say which substitution or which parts it used, and a rational function that needs partial fractions can come back unfinished, in which case the page says no antiderivative was found.
How do I write the expression?: Use the supported functions listed under What you can type. Multiplication can be left out where it is clear: 2x and 3 sin(x) both work, and the stage shows how the expression was read. A letter other than the integration variable is held as a constant, so x*y with respect to x gives x²*y/2 + C.
Can it do double and triple integrals?: Yes. Switch Integrals to Double or Triple and each extra variable gets its own pair of bounds. The inner integral runs first, so its bounds may use the variables outside it, which is how a region rather than a box is described; the outer bounds take numbers and constants only, and the page refuses a bound that names a variable already integrated. With no bounds at all you get the iterated antiderivative, for example x²*y²/4 for x times y. A double or triple value is also sampled over its region: a figure the sampling contradicts is not shown, and a numerical value is cut to the digits the sampling confirms.
Can it work in polar, cylindrical or spherical coordinates?: Yes, with the extra factor typed into the integrand, because the page integrates exactly what is in the box and adds nothing. Use single letters, such as r for the radius, t for the angle θ and p for the angle φ, then multiply by r for polar or cylindrical coordinates and by r^2*sin(p) for spherical ones. With the integrand r, r from 0 to 1 and t from 0 to 2*pi, a double integral gives pi, the area of the unit disc.
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.
How do I do an integral on a TI-84?: Press MATH and choose 9:fnInt(, then give it the expression, the variable and the two bounds, as in fnInt(X^2,X,0,3), which returns 9. The TI-84 works the value out numerically and has no antiderivative function, so an indefinite integral is the part it cannot do. This page gives both, in any current browser on a phone, tablet or computer, with nothing to install.
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.