Integral Calculator
Find an antiderivative with the rule for each term shown and checked, evaluate a definite integral exactly where a formula exists and numerically where it does not, and see the area shaded, with the parts below the x-axis counted as negative.
How to Use
- Type the function to integrate, for example x^2, x*e^x or 1/(1+x^2). Use ^ for powers and brackets round function arguments: sin(x), ln(x), sqrt(x), e^(-x^2).
- Leave the bounds empty for the indefinite integral (an antiderivative + C), or fill in a and b, such as 0 and 2*pi, for the definite integral from a to b.
- Read the antiderivative, the definite integral (exact when a formula exists, otherwise a decimal) and the total area.
- Show Work names the rule used for each term, checks the answer by differentiating it, and splits the interval where the curve crosses the x-axis.
- On the graph, area above the x-axis is shaded in the accent colour and counts as positive; area below is shaded red and counts as negative.
Worked Example
∫ x² dx from 0 to 2. By the power rule an antiderivative is F(x) = x³/3. Then F(2) − F(0) = 8/3 − 0 = 8/3, about 2.667: the area under the parabola between x = 0 and x = 2.
By parts: ∫ x·eˣ dx. Take u = x and dv = eˣ dx, so du = dx and v = eˣ. Then ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C. From 0 to 1: (e − e) − (0 − 1) = 1.
The common mistake: ∫ 1/x² dx from −1 to 1 “= −2”. Plugging the bounds into F(x) = −1/x gives −1 − 1 = −2, but 1/x² is positive everywhere, so its area cannot be negative. F(b) − F(a) only works when F is continuous on [a, b], and −1/x jumps to infinity at x = 0. The area near the spike is infinite, so the integral diverges.
Show Work
Formulas
From Exhaustion to the Fundamental Theorem
Archimedes, in the 3rd century BC, showed that a segment of a parabola has 4/3 the area of the triangle inscribed in it, by filling it with ever smaller triangles: the method of exhaustion. Isaac Barrow’s Lectiones Geometricae (1670) contained a geometric form of the link between areas and tangents, and Isaac Newton, who succeeded Barrow at Cambridge in 1669 and had worked on calculus since 1665–1666, and Gottfried Wilhelm Leibniz made it the main method of calculus. Leibniz first wrote the long S, ∫, for “summa”, in a manuscript of 29 October 1675, and published it in 1686.
Bernhard Riemann gave the definition of the integral as a limit of sums in 1854. Thomas Simpson published the rule now named after him in 1743, though it was known earlier, and Joseph Liouville showed in the 1830s that some integrals, such as that of e^(−x²), have no answer in elementary functions at all.
About This Tool
This calculator integrates symbolically with the same engine as the Calculus Workbench, adding term-by-term integration, u-substitution and a tidy-up of the answer, and it checks every antiderivative by differentiating it. It handles polynomials and powers, eˣ, sin, cos, tan, sec, sinh, cosh, ln, 1/(ax + b), 1/(1 + x²), a polynomial times eˣ, sin or cos (by parts), and integrands of the form c·g(u)·u′. When it finds no formula, as for e^(−x²) or sin²(x), the definite integral is found numerically by adaptive Simpson’s rule and labelled as numeric.
For a definite integral it splits the interval where the curve crosses the x-axis, to give the net and the total area, and where the function is unbounded, so it can tell a convergent improper integral such as ∫ 1/√x dx from 0 to 1 = 2 from a divergent one such as ∫ 1/x² dx from −1 to 1.
Everything runs in your browser; nothing is sent anywhere.
Related tools: Calculus Workbench, Derivative Calculator, and Numerical Methods Lab.
Frequently Asked Questions
What is the difference between net area and total area?
The definite integral adds the area above the x-axis and subtracts the area below it. For x³ − x from −1 to 2 the curve is above the axis on (−1, 0) and (1, 2) and below it on (0, 1): the three pieces are 1/4, −1/4 and 9/4, so the integral (the net area) is 9/4 = 2.25, while the total area is 1/4 + 1/4 + 9/4 = 11/4 = 2.75. For sin(x) from 0 to 2π the net area is 0 and the total area is 4.
Why is there a + C?
Differentiating removes constants, so every function has many antiderivatives that differ by a constant: x³/3, x³/3 + 5 and x³/3 − 2 all have derivative x². The + C stands for all of them. It cancels in a definite integral: F(2) − F(0) = 8/3 whichever C you choose.
Why can’t some integrals be done exactly?
Some simple functions have antiderivatives that cannot be written with ordinary functions at all; e^(−x²), sin(x)/x and √(1 + x³) are classic examples, and Joseph Liouville proved results of this kind in the 1830s. Their definite integrals still have values: ∫ e^(−x²) dx from 0 to 1 ≈ 0.7468241328, which this calculator finds with adaptive Simpson’s rule and labels as numeric.
What does it mean when an integral diverges?
That the area is infinite, usually because the function blows up inside the interval. ∫ 1/x² dx from −1 to 1 has an infinite spike at x = 0, so it diverges. Applying F(b) − F(a) with F(x) = −1/x anyway gives −1 − 1 = −2, a negative area for a function that is never negative. A spike does not always mean divergence: ∫ 1/√x dx from 0 to 1 = 2, because 2√x has a finite limit at 0.
Which integration rules does it use?
The power rule (∫ xⁿ dx = xⁿ⁺¹/(n + 1), with ∫ 1/x dx = ln|x|), the standard integrals of sin, cos, sec, tan, eˣ, sinh and cosh, ∫ 1/(1 + x²) dx = arctan(x), the reverse chain rule for inner functions such as 3x + 1, integration by parts for a polynomial times eˣ, sin or cos (∫ x·eˣ dx = x·eˣ − eˣ + C), and u-substitution, such as ∫ x/(x² + 1) dx = (1/2)ln(x² + 1) + C. Every antiderivative is checked by differentiating it back.
How do I use the Integral Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Homework check
∫ x·eˣ dx = x·eˣ − eˣ + C by parts, and exactly 1 from 0 to 1.
Distance from speed
v(t) = 3t² m/s from t = 0 to 4 s covers ∫ 3t² dt = 64 m.
Net and total area
x³ − x on [−1, 2]: net area 9/4, total area 11/4.
Probability
∫ e^(−x²) dx from 0 to 1 ≈ 0.7468, with no formula; times 2/√π it gives erf(1) = 0.8427.
Work done
A spring with k = 200 N/m stretched 0.1 m stores ∫ 200x dx from 0 to 0.1 = 1 J.
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