Antiderivative

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. For example: F(x) = 1/3 x³ is an antiderivative of f(x) = x².

Antiderivatives are important because they can be used to compute integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

<math>\int_a^b f(x)\, dx = F(b) - F(a)</math>

Because of this connection, the set of all antiderivatives of a given function f is sometimes called the general integral or indefinite integral of f and is written as an integral without boundaries:

<math>\int f(x)\, dx</math>

If F is an antiderivative of f and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G(x) = F(x) + C for all x. C is called the arbitrary constant of integration.

Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary:

<math>F(x) = \int_a^x f(t)\,dt</math>

This is another formulation of the fundamental theorem of calculus.

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

<math>\int e^{x^2}\,dx,\qquad \int \frac{\sin(x)}{x}\,dx,\qquad \int\frac{1}{\ln x}\,dx</math>

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.