Antiderivative

An antiderivative or indefinite integral is a mathematical function that is in the differential calculus, a branch of analysis examined.

Definition

Under one common function of a real function is defined as a differentiable function whose derivative function with matches. So is defined on an interval, it must be defined and differentiable on, and it must apply to any number of:

Existence and uniqueness

Every continuous on an interval function has an antiderivative. After the main theorem of differential and integral calculus is in fact integrable and the integral function

Is an antiderivative of.

Although integrated, but not all constant, then there is the integral function, it is not required at the locations where it is not continuous, is not to be differentiated, so it is generally not a primitive. It is necessary for the existence of a primitive function that the function satisfies the intermediate value theorem. This follows from the intermediate value theorem for derivatives.

Has a function, a primitive function, so it has even infinitely many. Indeed, if a primitive function of, then for any real number, the function defined by a primitive function of. If the domain of an interval, we obtain in this way all of the common functions: are two trunk and functions of, so is constant. Is the domain from no interval, the difference between two of primitives is not necessarily constant, but locally constant, i.e., constant on each contiguous subset of the domain.

Indefinite Integral

The term of the indefinite integral is not used consistently in the literature. For one, the indefinite integral of is understood as a synonym for a primitive function. The problem with this definition is that the term is absurd. Because in this case is the indefinite integral no picture because is not clear which of the infinitely many regular functions, the function to be imaged. Since the constant to differ all master functions, but often does not matter, this definition of the indefinite integral is only slightly problematic.

Another way to understand the indefinite integral, it is to define the term as the set of all primitive functions. This definition has the advantage that the indefinite integral linear mapping is analogous to the definite integral, although their values ​​are equivalence classes.

A more common way to define the indefinite integral, it is, it as a parameter integral

Conceive. Due to the fundamental theorem of differential and integral calculus, this expression gives for every continuous function is a function of strain. If we extend this definition still on Lebesgue integrals over arbitrary measure spaces, so the indefinite integral is generally not more primitive.

Examples

• An antiderivative of the polynomial is. The constant has been freely chosen, in this case, this primitive could be gained by reversing elementary derivation rules.
• If we consider the function then applies. However, no primitive of, because is not differentiable.

Application

Is a continuous on the compact, ie finite and closed interval (or more generally Riemann - integrable ) function, it can be using any primitive of the definite integral of over calculate:

Primitives are therefore needed for various calculations, eg:

• For determining the size of an area that is limited function graphs
• Volume calculation for rotating bodies

Seclusion / integration rules

For differentiating there are simple rules. In contrast, the situation with the indefinite Integrate is quite different, because the operation of the indefinite integrating leads to an expansion given Funktionensklassen, such as the integration is within the class of rational functions not completed and lists the functions and. Also, the class of so-called elementary functions is not complete. So Joseph Liouville proved that the simple function has no elementary antiderivative. The simple function has no elementary antiderivative. Contrast. The technique of integrating based on the following rules of integration:

• Additivity,
• The table of so-called basic integrals ( overview of the main functions and their primitives ),
• Method of partial integration,
• Substitution rule,
• Special procedures and
• Utilization of refined decompositions and transformations (eg polynomial, partial fractions, functional equations ).

Since there is no general rule for the determination of primitives, primitive functions are tabulated in so-called integral tables. Computer Algebra Systems ( CAS) are now able to calculate almost all previously tabulated integrals.

Primitives for complex functions

The term of the master function can be formulated also for complex functions. Because the derivative of a holomorphic function is holomorphic again, only holomorphic functions have antiderivatives. Holomorphy is already sufficiently locally: Is an area and a holomorphic function, then there is a neighborhood of in and a primitive function of, ie, for everyone.

The question of the existence of common features on all related to topological properties of together. For a holomorphic function, the following statements are equivalent:

For a region are equivalent: