Integral Calculator

Integration answers two questions that turn out to be one: which function has a given derivative (the antiderivative), and how much a quantity accumulates over an interval (the definite integral, a net signed area). The fundamental theorem of calculus connects them, and this integral calculator uses both: it builds an antiderivative with steps, verifies it, and evaluates definite integrals exactly and numerically.

What an integral calculator computes

Any two antiderivatives of a function differ by a constant, so an indefinite integral carries a constant C, which cancels again in a definite integral. These are the rules the calculator relies on most:

The second line, the fundamental theorem of calculus, requires f to be continuous on [a, b], so the calculator scans the interval for vertical asymptotes before it trusts F(b) − F(a). Integration by parts comes from the product rule and handles products such as x·e^x; substitution undoes the chain rule. The integrator tries its rules in a fixed order and keeps the first chain that works.

Worked example 1: integration by parts on x·sin(x)

The product x·sin(x) is not in any table, so the calculator integrates by parts with u = x, which gets simpler when differentiated, and dv = sin(x) dx. Then du = dx and v = −cos(x):

Differentiating checks it: sin(x) gives cos(x), and the product rule turns −x·cos(x) into −cos(x) + x·sin(x), so only x·sin(x) remains. For the definite integral from 0 to π, F(π) = sin(π) − π·cos(π) = π and F(0) = 0, so the result is exactly π, about 3.141592654. Adaptive Simpson quadrature reaches the same value without the formula. This is the default example above.

Worked example 2: net signed area of a polynomial

For the integral of 3x^2 + 2x − 5 from 0 to 2, the power rule works term by term: 3·x^3/3 + 2·x^2/2 − 5x = x^3 + x^2 − 5x. That equals 8 + 4 − 10 = 2 at x = 2 and 0 at x = 0, so the integral is 2.

That 2 is a net value. The polynomial is zero at x = 1, negative before and positive after, so the piece from 0 to 1 contributes −3 and the piece from 1 to 2 contributes 5. They net to 2, while the total area between the curve and the axis is 3 + 5 = 8.

Substitution and partial fractions, step by step

When one factor is the derivative of an inner function, substitution collapses the integral. In ∫ x·e^(x^2) dx, u = x^2 and du = 2x dx, so the integral becomes ∫ e^u/2 du and the answer is e^(x^2)/2 + C. The calculator finds such substitutions by testing whether the remaining factors are a multiple of an inner expression's derivative.

Rational functions are split into partial fractions. For (2x + 3)/(x^2 + 3x + 2), the denominator factors as (x + 1)(x + 2) and the fraction equals 1/(x + 1) + 1/(x + 2), so the antiderivative is ln|x + 1| + ln|x + 2| + C and the integral from 0 to 1 is ln(3), about 1.098612289. Long division comes first when the numerator's degree is at least the denominator's, and an irreducible quadratic gives an arctangent, as in ∫ 1/(x^2 + 2x + 5) dx = arctan((x + 1)/2)/2 + C.

Exact values of common integrals on [0, 1]

Each antiderivative is the calculator's own output, and each value was computed as F(1) − F(0) and confirmed by quadrature.

Two rows share π/4: the arctangent integral, and a quarter of the unit circle, whose area is π/4 by geometry.

Riemann sums, the trapezoidal rule and Simpson's rule

Numerical rules approximate a definite integral from function values alone. With n subintervals of width Δx = (b − a)/n and points x0, x1, ..., xn:

For x^2 on [0, 2] with n = 4, the left sum is 1.75, the right sum 3.75, the midpoint rule 2.625, the trapezoidal rule 2.75 and Simpson's rule 2.666666667, against the exact 8/3. Simpson's rule is exact here because it is exact for every polynomial of degree three or less. The weights are the multipliers inside the brackets:

The next table integrates 1/x from 1 to 2, exactly ln(2) or about 0.6931471806. Each time n doubles, the trapezoidal error falls by a factor near 4 (3.92, 3.98, 3.99, 4.00) and Simpson's by a factor approaching 16 (12.15, 14.53, 15.56, 15.89), as the error terms in the NIST Digital Library of Mathematical Functions predict: −((b − a)/12)·h^2·f″(ξ) for the trapezoidal rule and −((b − a)/180)·h^4·f⁽⁴⁾(ξ) for Simpson's rule.

Errors are rule value minus ln(2). Both are positive: the trapezoidal rule overestimates because 1/x curves upward, Simpson's rule because the fourth derivative of 1/x is positive. Simpson's rule with n = 32 is correct to seven decimal places; the trapezoidal rule needs 1,450 subintervals for the same accuracy.

Double integrals and the order of integration

A double integral adds up f(x, y) over a region, one variable at a time. For x^2 + y^2 with x from 0 to 1 and y from 0 to 2, the inner integral in x is 1/3 + y^2, and the outer integral of that from 0 to 2 is 2/3 + 8/3 = 10/3, about 3.333333333, the default in the double integral mode. On a rectangle the order does not change the value of a continuous integrand, but for other regions it decides how the limits are written. To integrate x·y over the triangle under y = x for x from 0 to 1, use dy dx with inner limits 0 and x: the inner integral is x·x^2/2 and the outer integral is 1/8. With f = 1 and inner limits x^2 to x, the result is the area between the parabola and the line, 1/6. The calculator uses composite Simpson's rule on a 128 by 128 grid and estimates the error from coarser grids.

Improper integrals: infinite limits and vertical asymptotes

An integral is improper when a limit is infinite or the integrand blows up at an endpoint; it converges only if the defining limit is finite. Type inf or -inf for an infinite limit; endpoint singularities are detected automatically. The integral of 1/sqrt(x) from 0 to 1 is 2, of ln(x) from 0 to 1 is −1, of 1/x^2 from 1 to ∞ is 1, and of e^(−x) from 0 to ∞ is 1. The integral of 1/x from 1 to ∞ diverges because ln(x) grows without bound, and the calculator says so.

An asymptote inside the interval is a trap: applying F(b) − F(a) to the integral of 1/x^2 from −1 to 1 gives −2, impossible for a positive integrand. The calculator finds the asymptote at x = 0 and reports no value; split the integral there, and both pieces diverge.

How to use the calculator

Pick a mode. In the integral mode, type the function with ^ for powers, * or a space for multiplication, and names such as sqrt, e^x, ln, sin, cos, tan, sec, arcsin, arctan and abs, then choose indefinite or definite; the definite setting makes it a definite integral calculator with F(b), F(a) and a numerical cross-check. As on handheld calculators, ln(x) is the natural logarithm and log(x) is base 10. Use a single letter as the variable and put * between letters (x*y, not xy). Limits accept pi/2, sqrt(2) or inf. The result card shows how your input was read, then come the steps and a check table. The Riemann mode adds the number of subintervals n, compares five rules with the exact value and lists every sample point. The double integral mode is a double integral calculator for f(x, y): it takes the function, the order of integration and four limits.

Who it is for. Calculus I and II students, including AP Calculus students, use the steps to check homework; the step titles match textbook names. Engineering and physics students compute work, charge and centers of mass, and apply the numerical rules to measured data. Statistics students integrate probability densities, where an area is a probability.

Key terms

• Integrand: the function being integrated.
• Antiderivative: a function F with F′ = f, also called the indefinite integral.
• Definite integral: F(b) − F(a), the net signed area from a to b.
• Riemann sum: an approximation of the area by rectangles.
• Improper integral: an integral with an infinite limit or an unbounded integrand, defined as a limit.
• Iterated integral: a double integral evaluated one variable at a time.
• Elementary function: a function built from polynomials, roots, exponentials, logarithms and trig functions.

What changes the result, and how to check it

Swapping the limits flips the sign. Correct antiderivatives can look different: sin^2(x)/2, −cos^2(x)/2 and −cos(2x)/4 all integrate sin(x)·cos(x), because they differ by constants. ln|x| covers both sides of zero, while ln(x) covers only x > 0, and trig functions use radians. To check a result by hand, differentiate it, and for a definite integral compare it with a rough sketch of the area; a wrong sign or size points to an algebra slip or a missed asymptote.

Limitations of a rule-based integral solver

"No closed form found" means the rules ran out, not that no antiderivative exists. Some integrals, like e^(x^2), sin(x)/x and 1/ln(x), provably have no elementary antiderivative; others, like 1/(x^4 + 1), need techniques beyond these rules. Definite integrals still get a numerical value. Oscillating integrals over infinite intervals such as sin(x)/x from 0 to ∞ are reported without a value, and the double integral mode is numerical only.

Related calculators

• To differentiate a function step by step, use the Derivative Calculator.
• For limits, including those behind improper integrals, use the Limit Calculator.
• To replace a function by a polynomial first, use the Taylor Series Calculator.
• To see the area you are integrating, use the Graphing Calculator.
• For areas and volumes of standard shapes, use the Area Calculator and the Volume Calculator.
• For the average slope between two points, use the Rate of Change Calculator.

Sources

The rules and error formulas are standard results; every antiderivative, value and table entry on this page was produced by the calculator itself.

• NIST Digital Library of Mathematical Functions, §1.4 Calculus of One Variable: integration by parts (1.4.16) and the fundamental theorem of calculus (1.4.26).
• NIST DLMF §1.2(iii) Partial Fractions.
• NIST DLMF §1.5(v) Multiple Integrals: change of order of integration (1.5.35, 1.5.36).
• NIST DLMF §3.5 Quadrature: trapezoidal rule (3.5.2, error 3.5.3) and Simpson's rule (3.5.7, error 3.5.8).
• NIST DLMF §4.10 and §4.26: integrals of logarithms, exponentials and trigonometric functions.
• mathjs derivative function, used to verify every antiderivative.