Does the improper integral converge?

How it works

The page evaluates ∫ from a to b of f(x) dx, where a may be −∞, b may be ∞, and f may be infinite at a or b. It says Converges with the value, or Diverges.

A computer algebra system (nerdamer, open source) does the algebra, in the background after you start typing. The work is split into pieces:

• Both limits finite: one definite integral from a to b, worked out as on the integral calculator: F(b) − F(a) with a checked antiderivative F. Where f is infinite at an end, the limit of F from inside the interval is used; if |F| keeps growing towards that end, the integral diverges. A removable gap inside (sin(x)/x at 0) changes nothing; a pole inside, or a part of [a, b] where f is not a real number, gives no value.
• b = ∞: split at c = a when a is more than 1 (as you typed it), else c = 1. From a to c (when a is less than c) it is a finite definite integral as above. From c to ∞ the substitution x = 1/t gives ∫ from 0 to 1/c of f(1/t)/t² dt, a definite integral with an improper end at t = 0.
• a = −∞: split at c = b when b is less than −1 (as you typed it), else c = −1. From c to b (when c is less than b) it is a finite definite integral; from −∞ to c the substitution x = −1/t gives ∫ from 0 to −1/c of f(−1/t)/t² dt.
• a = −∞ and b = ∞: three pieces, split at −1 and 1.

For a piece to ±∞ whose substituted integral has no exact form, the page also tries the antiderivative F of f: its limit L at ±∞ less F(c) (the other way round at −∞). This exact form is used when its value agrees with the piece's value to 1 part in 10⁹.

Divergence at ±∞. When the substituted integral of a piece has no answer, the page takes the antiderivative F of f and looks at |F(x)| at x = 8, 8², 8³, …, 8⁸ = 16,777,216 (their negatives at −∞), so a pole of f out to about 10⁶, where |F| grows and then falls, does not pass for divergence. If, from the third point on, each value is too large for a double or both grows from the point before and grows by at least 0.9 times the growth before that, F grows without bound and the integral diverges (1/x from 1 to ∞: F = ln|x| grows by ln 8 each time x grows 8 times). Otherwise the page gives no answer.

The value is the sum of the pieces. The exact value is shown when every piece has one: the pieces are added (with atan(1) written as π/4 and atan(−1) as −π/4) and simplified by the algebra when the simplified sum stays exact; otherwise the sum is shown as it is, unless it holds a rounded number, when only the decimal is shown.

Every piece is checked. Its value F(end) − F(start) must agree with a numeric integral of the function (Gauss-Kronrod quadrature, or tanh-sinh quadrature at an improper end) to 1 part in 10⁷ of the integral of its absolute value; the antiderivative is checked by differentiating it numerically at 20 points. A result that holds a number the algebra could only give rounded (a whole number past 2⁵³; a fraction p/q, p and q being the numbers that multiply its top and its bottom, such as 288557167/(342919925e), when q after removing its factors 2 and 5 is over 1,000,000, or is over 1 while |p| times it is over 10¹²; or a decimal with more than 12 significant digits) is never shown as exact.

What you can type

• A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
• The function f uses the variable x. Numbers can have decimals (2.5) and powers of ten (1e-3).
• Operations: + − * / and ^ for powers. Brackets group. A number or bracket next to a letter multiplies: 2x, 3(x + 1), x sin(x).
• Constants: pi (or π) and e.
• Functions: sqrt, cbrt, ln (and log, the same natural logarithm), log10, exp, abs (or |x|), sin, cos, tan, sec, csc, cot, asin, acos, atan (arcsin, arccos, arctan also work), sinh, cosh, tanh and their inverses. Angles are in radians. sin x without brackets means sin(x); sin x^2 means sin(x²).
• The limits a and b are numbers or constant expressions (0, 1, pi, e), or inf and -inf (also infinity, ∞ and oo). a must be less than b.

What gets no answer

• A pole strictly between a and b (1/x³ from −1 to 1, or 1/(x − 2·10⁶)² from 1 to ∞): split the integral there and enter the two parts. A pole past 8⁸ on an infinite interval can still show as Diverges (then the integral does diverge, as |F| grows without bound towards that pole).
• An integral that oscillates without settling (sin(x) from 0 to ∞), or grows too slowly to tell (1/(x ln²x) from 2 to ∞ converges, but its antiderivative approaches its limit too slowly for the checks).
• A step that fails its check, finds no formula (e^(−x²) from 0 to ∞ needs erf at ∞, which the algebra cannot take), or takes over 3 seconds.

Worked examples by hand

∫ from 1 to ∞ of π/x² dx (OpenStax Calculus Volume 2, section 3.7, Example 3.48: the volume of Gabriel's horn). An antiderivative is −π/x, which tends to 0 as x → ∞. The integral is 0 − (−π/1) = π: it converges.

∫ from −∞ to 0 of 1/(x² + 4) dx (OpenStax Calculus Volume 2, section 3.7, Example 3.50). An antiderivative is atan(x/2)/2, which tends to −π/4 as x → −∞. The integral is 0 − (−π/4) = π/4 = 0.7853981634: it converges. The page writes its exact value as atan(2)/2 − atan(−1/2)/2, which equals π/4 because atan(2) + atan(1/2) = π/2.

∫ from 0 to 4 of 1/√(4 − x) dx (OpenStax Calculus Volume 2, section 3.7, Example 3.52). f is infinite at 4. An antiderivative is −2√(4 − x), which tends to 0 as x → 4. The integral is 0 − (−2√4) = 4: it converges.

∫ from 1 to ∞ of 1/x dx (OpenStax Calculus Volume 2, section 3.7, Example 3.47). An antiderivative is ln|x|, which grows without bound: the integral diverges.