Question 1

How do I calculate a definite integral?

Accepted Answer

The definite integral of f from a to b is the signed area between the graph and the x-axis. If F is an antiderivative of f, the fundamental theorem of calculus gives ∫ₐᵇ f(x) dx = F(b) − F(a). For ∫ from −2 to 2 of (x² − 4) dx, F(x) = x³/3 − 4x and the answer is −32/3 (OpenStax Calculus Volume 1, Example 5.20). This calculator does not depend on finding F: it integrates numerically with an adaptive Gauss–Kronrod rule, shows the estimated error, and when it also finds an antiderivative it shows the exact value and confirms that the two agree.

Question 2

Which antiderivatives can it find?

Accepted Answer

Sums and constant multiples, powers (xⁿ with any real n, with 1/x giving ln|x|), eˣ and aˣ, sin, cos, tan, cot, sec, csc, sec², csc², sec·tan, csc·cot, 1/(1 + x²), 1/√(1 − x²), ln x and the inverse trigonometric functions, with a linear argument f(ax + b) and with a general substitution u = g(x) when g′(x) is in the integrand (∫6x(3x² + 4)⁴ dx = (3x² + 4)⁵ ÷ 5, OpenStax Example 5.30). Quotients such as (x² + 4∛x) ÷ x are split into powers first, quadratics under a root or in a denominator are handled by completing the square, and polynomial × eˣ, sin x, cos x or ln x is integrated by parts. Anything else gets the honest answer “no antiderivative found by these rules” — for example eˣ², sin(x²) or sin(x)/x, which have no elementary antiderivative at all, and integrals that need partial fractions or trigonometric substitution. A definite integral of such a function is still computed numerically.

Question 3

How accurate is the number, and what is the error estimate?

Accepted Answer

The integral is computed with the 15-point Kronrod rule and the embedded 7-point Gauss rule from QUADPACK; the difference between the two is the error estimate, and the interval with the largest error is halved until the estimated error is below about 10⁻¹⁰ of the value. For smooth functions the result is correct to 12 or more digits. The estimate is not a guarantee, and sharp spikes between the sample points can fool any numerical method. Integrable singularities at an end point are fine: ∫₀¹ 1/√x dx = 2 and ∫₀¹ ln x dx = −1 are computed correctly. If the error will not come down, the integral may diverge (∫₀¹ 1/x dx) and the page says the integration did not settle instead of printing a number.

Question 4

Can it do improper integrals with infinite limits?

Accepted Answer

Yes: type inf or -inf as a limit. An infinite interval is mapped onto a finite one by a change of variable (x = a + 1/s − 1 with s between 0 and 1 for [a, ∞); the whole line is split at 0 into two such pieces) and integrated in the same way, so a singular end point that results from it is handled by the same interval halving. Checks: the Gaussian ∫ from −∞ to ∞ of e^(−x²) dx gives √π ≈ 1.7724538509, ∫₀^∞ e^(−x) dx = 1, ∫₁^∞ 1/x² dx = 1 and ∫ from −∞ to ∞ of 1/(1 + x²) dx = π. Convergence is not proved: for ∫₁^∞ 1/x dx, which diverges, the method breaks down and the page tells you so.

Question 5

What does the constant C mean?

Accepted Answer

An antiderivative is only defined up to a constant, because the derivative of a constant is 0. ∫x² dx = x³/3 + C, where C is any number. The calculator verifies every antiderivative it reports by differentiating it with its own derivative code and comparing with your integrand at a few dozen test points, and it shows the derivative so you can see that it returns the integrand (OpenStax §4.10, “Verifying an Indefinite Integral”). Definite integrals do not need C: it cancels in F(b) − F(a).

Question 6

How do I type the integrand and the limits?

Accepted Answer

Use x as the variable, ^ for powers and write products as 2x or x*sin(x); functions are sin, cos, tan, sec, csc, cot, arcsin, arctan, sinh, cosh, ln, log (base 10), log_2(x), sqrt, exp and abs, with the constants pi and e. The limits can be numbers or expressions such as pi/2, sqrt(3)/3, ln(2) or 2*pi, and inf for infinity. Angles are in radians. The page shows how it read your input, because 1/2x means (1/2)·x; write 1/(2x) for the other reading.

Question 7

Is the answer the area? What about negative values?

Accepted Answer

The integral is the signed area: parts below the x-axis count as negative. ∫ from −3 to 3 of 2x dx is 0 because two triangles of area 9 cancel (OpenStax Example 5.9). For the total area between the graph and the axis integrate the absolute value: ∫₀³ |3x − 5| dx = 41/6 ≈ 6.8333, while ∫₀³ (3x − 5) dx = −3/2 (Examples 5.24 and 5.25). Type abs(3x-5) to get it.

Question 8

What are the most common mistakes when integrating?

Accepted Answer

Forgetting + C in an indefinite integral; integrating a product factor by factor (the product of antiderivatives is not an antiderivative of the product: x·eˣ has antiderivative xeˣ − eˣ, not x²eˣ/2, as OpenStax shows in Example 4.51); using the power rule on 1/x (the answer is ln|x|, not x⁰/0); mixing up the order of the limits, which changes the sign of a definite integral; and typing angles in degrees, because the calculator and calculus use radians. When you can’t tell whether your own antiderivative is right, differentiate it: you must get the integrand back, which is exactly the check the calculator performs.

Integral calculator