Indefinite integrals

The app integrates with the methods taught at school and says at every step what it does. It checks the result by differentiating (the check is the last step) and draws the integrand and the antiderivative.

Roots of powers of the variable (∛(x⁴), √(x³)) are rewritten as powers with rational exponents; an absolute value (|cos x|, √(1 − sin²x)) is written with the sign function sgn.

If an integral has no elementary antiderivative (such as e^(−x²)), the app says so and gives it in terms of a special function (erf, Si, …).

How to enter it

int(x e^x, x)An integral with the variable.
int(x^2 sin x)Without the variable when there is only one.
∫ x^2 dxWith the integral sign, also from a photo.
x e^xA function with the “Integrate” chip.

Basic integrals and substitution

The integral of a sum term by term, constants taken out, table integrals (also with a linear inner function) and substitution when the integrand contains a function together with the derivative of its inside.

Examples

Term by term from the table.
Input: int(3x^2 - 2/x + e^x)
int(3x^2 - 2/x + e^x)
int(3x^2 - 2/x + e^x)int(3x^2 - 2/x + e^x)
The substitution t = x².
Input: int(2x cos(x^2))
int(2x cos(x^2))
int(2x cos(x^2))int(2x cos(x^2))
An integral with constants a, b.
Input: int(1/(a - bx)^3)
int(1/(a - bx)^3)

Integration by parts

The app integrates a product of a polynomial with an exponential, a trigonometric function or a logarithm by parts, repeatedly if needed; integrals like eˣ sin x by cyclic integration by parts.

Examples

Integration by parts twice in a row.
Input: int(x^2 e^x)
int(x^2 e^x)
int(x^2 e^x)int(x^2 e^x)

Rational functions

The app divides a rational function if needed, factors the denominator and splits the function into partial fractions, which it integrates into logarithms and arctangents.

A denominator that can’t be factored over the rationals (x⁴ + 1, x⁵ + 1) is factored over the reals – each pair of complex conjugate roots gives a quadratic factor with no real roots.

Examples

Partial fractions.
Input: int((x + 1)/(x^2 - 3x + 2))
int((x + 1)/(x^2 - 3x + 2))
int((x + 1)/(x^2 - 3x + 2))int((x + 1)/(x^2 - 3x + 2))
Factoring the denominator x⁴ + 1 over the reals.
Input: int(1/(x^4 + 1))
int(1/(x^4 + 1))
int(1/(x^4 + 1))int(1/(x^4 + 1))int(1/(x^4 + 1))

Trigonometric functions

Identities first (sin²x = (1 − cos 2x)/2, products into sums), for an odd power the substitution t = sin x or t = cos x, for a rational function of sine and cosine a substitution by parity, and the Weierstrass substitution t = tan(x/2) only as the last resort.

Examples

An odd power of sine – the substitution t = cos x.
Input: int(sin(x)^3 cos(x)^2)
int(sin(x)^3 cos(x)^2)
int(sin(x)^3 cos(x)^2)int(sin(x)^3 cos(x)^2)

Hyperbolic functions

Table integrals (∫sinh x dx = cosh x + C, ∫1/cosh²x dx = tanh x + C, …), ∫tanh x dx by the logarithm of the derivative, by parts (x sinh x, arsinh x), squares by the formulas sinh²x = (cosh 2x − 1)/2, tanh²x = 1 − 1/cosh²x; the rest by the definition through exponentials.

Examples

The formula sinh²x = (cosh 2x − 1)/2.
Input: int(sinh(x)^2)
int(sinh(x)^2)

Roots

Integrals with a root of a linear fractional function by substitution, with a root of a quadratic by an Euler substitution, and binomial integrals by Chebyshev's conditions – when none of them holds, the app says the integral is not elementary.

Examples

An Euler substitution.
Input: int(sqrt(x^2 + 1))
int(sqrt(x^2 + 1))
int(sqrt(x^2 + 1))int(sqrt(x^2 + 1))int(sqrt(x^2 + 1))

Known limitations