In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i.e. F ' = f. The process of finding antiderivatives is antidifferentiation (or indefinite integration).
For example: F(x) = x³ / 3 is an antiderivative of f(x) = x². As the derivative of a constant is zero, x² will have an infinite number of antiderivatives; such as (x³ / 3) + 0 and (x³ / 3) + 7 and (x³ / 3) - 36...thus; the antiderivative family of x² is collectively referred to by F(x) = (x³ / 3) + C; where C is any constant. Essentially, related antiderivatives are vertical translations of each other; each graph's location depending upon the value of C.
Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary:
There are also some non-continuous functions which have an antiderivative, for example f(x) = 2x sin (1/x) - cos(1/x) with f(0) = 0 is not continuous at x = 0 but has the antiderivative F(x) = x² sin(1/x) with F(0) = 0.
There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations). Examples of these are
Techniques of integration
Finding antiderivatives is considerably harder than finding derivatives. We have various methods at our disposal:
- the linearity of integration allows us to break complicated integrals into simpler ones,
- integration by substitution, often combined with trigonometric identities
- integration by parts to integrate products of functions,
- the inverse chain rule method, a special case of integration by substitution
- the method of partial fractions in integration allows us to integrate all rational functions (fractions of two polynomials),
- the natural logarithm integral condition,
- the Risch algorithm,
- integrals can also be looked up in a table of integrals.
- When integrating multiple time, we can use certain additional techniques, see for instance double integrals and polar co-ordinates, the Jacobian and the Stokes theorem.
- If a function has no elementary antiderivative (for instance, exp(x²)), an area integral can be approximated using numerical integration.