number such that for all . is called the constant of integration. If the domain of is a disjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance is the most general antiderivative of on its natural domain Every continuous function has an antiderivative, and one antiderivative is given by the definite integral of with variable upper boundary: Varying the lower boundary produces other antiderivatives (but not necessarily all possible antiderivatives). This is another formulation of the fundamental theorem of calculus. 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 the error function the Fresnel function the sine integral the logarithmic integral function and sophomore's dream For a more detailed discussion, see also Differential Galois theory. Techniques of integration Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals). For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. To learn more, see elementary functions and nonelementary integral. There exist many properties and techniques for finding antiderivatives. These include, among others: The linearity of integration (which breaks complicated integrals into simpler ones) Integration by substitution, often combined with trigonometric identities or the natural logarithm The inverse chain rule method (a special case of integration by substitution) Integration by parts (to integrate products of functions) Inverse function integration (a formula that expresses the antiderivative of the inverse of an invertible and continuous function , in terms of the antiderivative of and of ). The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials) The Risch algorithm Additional techniques for multiple integrations (see for instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem) Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of ) Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used) Cauchy formula for repeated integration (to calculate the -times antiderivative of a function) Computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques