How the integral calculator works
Integration reverses differentiation, but unlike differentiation there is no single mechanical procedure that always works. The calculator therefore tries techniques in order, the same order a student would: split sums, pull out constants, look the integrand up in the standard table, then try substitution, integration by parts, and partial fractions. Each successful rule is written down as a step. If no technique applies, the calculator says so rather than guessing, and falls back to a numeric method when bounds are supplied.
Worked example: the integral of x e^x
The integrand is a product of a polynomial and an exponential, which is the signature case for integration by parts. Choose u = x (it gets simpler when differentiated) and dv = e^x dx (it is easy to integrate). Then du = dx and v = e^x. The parts formula u v - the integral of v du gives x e^x - the integral of e^x dx = x e^x - e^x. Adding the constant, the answer is x e^x - e^x + C, which the calculator writes as -e^x + x e^x + C. Differentiate it to check: e^x + x e^x - e^x = x e^x.
Table of basic integrals
| Integrand | Antiderivative | Integrand | Antiderivative |
|---|---|---|---|
| k | k x | sec^2 x | tan x |
| x^n (n not -1) | x^(n+1) / (n+1) | sec x tan x | sec x |
| 1 / x | ln |x| | csc^2 x | -cot x |
| e^x | e^x | csc x cot x | -csc x |
| a^x | a^x / ln a | 1 / (1 + x^2) | atan x |
| sin x | -cos x | 1 / sqrt(1 - x^2) | asin x |
| cos x | sin x | ln x | x ln x - x |
| tan x | -ln |cos x| | sinh x, cosh x | cosh x, sinh x |
Techniques and when they apply
| Technique | Recognise it by | Example |
|---|---|---|
| Linear substitution | A table function of ax + b | sin(3x) gives -cos(3x) / 3 |
| u-substitution | An inner expression whose derivative is a factor | 2x e^(x^2) gives e^(x^2) |
| Integration by parts | x^n times e^(ax), sin(ax), cos(ax), or ln x | x sin x gives sin x - x cos x |
| Partial fractions | A rational function whose denominator factors | 1 / ((x - 1)(x - 2)) |
| Completing the square | 1 / (quadratic) with no real roots | 1 / (x^2 + 2x + 5) |
| Expansion | A product of polynomials | (x^2 + 1)^2 |
Definite integrals
With both bounds filled in, the calculator first looks for an antiderivative F and reports F(b) - F(a) with the two values shown, which is the fundamental theorem of calculus in action. When F cannot be found, or F is infinite at a bound, it switches to adaptive Simpson quadrature: the interval is split recursively until each piece's parabola estimate is accurate to the requested tolerance. The result is then labelled numeric so you know it is an approximation rather than a closed form.
Reading the steps
Each step names the rule and shows the integral it applies to on the left. Integrals still to be done appear as placeholders of the form ∫[...] dx, and later steps resolve them. Substitution steps show the new variable and its differential, then a Substitute back step returns to the original variable. The final Simplify step merges terms and adds the constant of integration.
What the calculator cannot do
It does not know trigonometric substitution for sqrt(x^2 + a^2) and its relatives, repeated linear factors in partial fractions such as 1 / (x (x + 1)^2), or products of three or more transcendental functions. For those, supply bounds to get a numeric value, or split the problem by hand and integrate the pieces.