Integrals

Integrals computes the antiderivative of a function, its definite integral between two bounds, and the area under the curve — with the full workings, step by step. WHAT IT DOES • The antiderivative, in exact form, with its constant of integration. • The definite integral between two bounds, which accept an expression: 0, pi/2, e, 2*pi… • The curve of the function and of its antiderivative, with the area shaded between the bounds. That picture is what makes the result obvious. • The workings, where every step is announced by the rule it uses: linearity, standard antiderivative, u′/u form, substitution, partial fractions, trigonometric substitution, integration by parts. IT CAN FAIL, IT CANNOT ANSWER WRONG Differentiating is mechanical; integrating is not. There is no single rule to apply: it is a search, where you try a path and back out of it. So every antiderivative found is differentiated again, then compared numerically to the original function, before being shown. A path that yields an unverifiable result is dropped, and the app honestly says it does not know — which happens, in particular for functions that have no elementary antiderivative at all, such as e^(x²), sin(x)/x or ln(ln x). WHAT IT UNDERSTANDS Polynomials and rational fractions, roots, exponentials and logarithms, trigonometric and hyperbolic functions and their inverses, and compositions of all these. When no antiderivative exists, the definite integral is still computed numerically, by Simpson quadrature. DARK MODE The app follows the system theme, formulas and curves included. PRO VERSION The step-by-step workings unlock by watching a short ad, or for good with the Pro version — a single payment, which also removes the ads. Integrals is the counterpart of Derivatives, by the same author: both apps share the same symbolic engine.