The Indefinite Integral. The Concept of an Antiderivative, Basic Concepts and Definitions

Definition

Antiderivative (sometimes also called a primitive function or primitive) is one of the most important concepts of mathematical analysis of a real variable (there are also generalizations of this concept for complex functions).

A function is called an antiderivative of a function on an interval , finite or infinite, if the function is differentiable at every point of this interval and its derivative satisfies the following equality:

The last equality can be written in terms of differentials:

or

An antiderivative of a given function is a function whose derivative is equal to (on the entire domain of ), that is, . Finding an antiderivative is the operation inverse to differentiation: differentiation finds the derivative of a given function, whereas by finding an antiderivative we, conversely, recover the original function from its given derivative.

Antiderivatives are important because they allow us to compute definite integrals. If is an antiderivative of an integrable continuous function , then:

This relation is called the Newton–Leibniz formula.

Technically, finding an antiderivative amounts to computing the indefinite integral of , and the process itself is called integration. For applications of this theory in geometry, see Integral calculus.

Example: the function is an antiderivative of because

Example

The function is an antiderivative of the function , since

The antiderivative has a finite derivative and is therefore a continuous function.

Theorem

(On the infinite set of antiderivatives of a function)

If a function is an antiderivative of a function on some interval, then the function , where is an arbitrary constant, is also an antiderivative of the function on the interval under consideration.

Example

It is known that an antiderivative of the function is the function , and therefore all functions of the form are also antiderivatives, since the equality holds:

Thus, if a function has an antiderivative, then it has infinitely many antiderivatives.

Theorem

(On the general form of an antiderivative of a function)

If the functions and are any two antiderivatives of the function , then their difference is equal to some constant, that is,

The last theorem can be stated differently: every function that is an antiderivative of the function can be represented in the form .

Properties of the Antiderivative

• An antiderivative of a sum is the sum of the antiderivatives.
• An antiderivative of the product of a constant and a function is the product of the constant and an antiderivative of the function.
• Every function that is continuous on a closed interval has both an antiderivative and a Riemann integral. In general, however, the existence of an antiderivative and the integrability of a function are unrelated :
  • The sign function (sgn) is Riemann integrable but has no antiderivative (because of the discontinuity at zero).
  • The function (with also set) has a finite derivative on the interval , so the function has an antiderivative (namely, ), but is unbounded on and therefore is not Riemann integrable.

The Indefinite Integral

Definition

The set of all antiderivatives of a function defined on a given interval is called the indefinite integral of the function and is denoted by the symbol . That is,

The sign is called the integral sign, is the integrand expression, is the integrand, and is the variable of integration.

The operation of finding an antiderivative or the indefinite integral of a function is called integration of the function . Integration is the operation inverse to differentiation.

Geometric Interpretation of the Indefinite Integral

The indefinite integral is a family of parallel curves , where each specific numerical value of the constant corresponds to a particular curve of this family.

The graph of each curve in the family is called an integral curve.

Theorem

Every function that is continuous on an interval has an antiderivative on this interval.

Properties of the Indefinite Integral

1. The differential of an indefinite integral is equal to the integrand expression

Example

2. The derivative of an indefinite integral is equal to the integrand

Example

3. The indefinite integral of the differential of a function is equal to that function plus an arbitrary constant

Example

4. A constant factor can be taken out of the indefinite integral sign or brought under the integral sign

Example

5. The indefinite integral of the sum/difference of two or more functions is equal to the sum/difference of the indefinite integrals of those functions

Example

6. If , then as well, where the function is an arbitrary function with a continuous derivative.

Example

It is known that , and then

Integration Techniques Methods of Integration List of Integrals of Elementary Functions

Finding antiderivatives is considerably harder than finding derivatives. There are several methods for it:

• the linearity of integration allows complex integrals to be broken into parts,
• integration by substitution, often used together with trigonometric identities or the natural logarithm,
• integration by parts for handling products of functions,
• the reverse chain rule method, a special case of integration by parts,
• the method of integrating rational fractions makes it possible to integrate any rational function (fractions with polynomials in the numerator and denominator),
• the Risch algorithm, an algorithm for integrating any elementary functions,
• some integrals can be found in a table of integrals,
• for multiple integration, additional techniques can be used; for example, see the double integral and polar coordinates, the Jacobian and Stokes' theorem,
• computer algebra systems help automate some of the symbolic operations above (in particular the Risch algorithm), which is very convenient when algebraic computations become too cumbersome,
• if a function has no elementary antiderivative (such as, for example, ), its integral can be computed approximately using numerical integration.

Table of Integrals, Table of Indefinite Integrals

In the table of integrals we have tried to gather the most complete collection of formulas, which will help you solve any integral. The constant companions of the table of integrals are the table of derivatives and the derivative formulas, which are also presented in full on our website.

Basic Formulas

Methods for Solving Indefinite Integrals

1. The Method of Direct Integration

Reduction to tabular form or the method of direct integration. Using identity transformations of the integrand, the integral is reduced to one to which the basic rules of integration apply and the table of basic integrals can be used.

Example

Problem. Find the integral

Solution. We will use the properties of the integral and reduce the given integral to tabular form.

Answer.

The Method of Direct Integration

Definition

An integration method in which, by means of identity transformations of the integrand and application of the properties of the integral, the integral is reduced to one or several tabular integrals is called direct integration.

Thus, the algorithm is as follows:

1. identity transformation of the integrand;
2. application of the properties of the indefinite integral: taking the constant out of the integral sign, and representing the integral of a sum of functions as a sum of integrals;
3. use of the table of integrals.

In the simplest examples, to apply direct integration it is enough to decompose the integrand into terms and take the constant quantities out of the integral sign.

With some practice in integration, these steps are usually carried out mentally, writing down only the result of the integration.

2. Introducing a Function Under the Differential Sign

In the formula for the indefinite integral, the quantity means that the differential of the variable is taken. Some properties of the differential can be used to complicate the expression under the differential sign and thereby simplify finding the integral itself. The following formula is used for this

If the required function is absent, it can sometimes be produced by algebraic transformations.

Integration by Introducing a Function Under the Differential

Suppose we need to find the indefinite integral . Assume that there exist differentiable functions and such that

Then

This transformation of the integrand expression is called introducing a function under the differential sign.

Then, if and , the following equality holds:

Remark. When integrating by introducing a function under the differential sign, the following differential equalities are useful:

Example

Problem. Using the introduction of a function under the differential, find the indefinite integral

Solution. We introduce under the differential sign, thereby reducing the original integral to a tabular one.

Answer.

In general form, the following equality holds:

Example

Problem. Find the integral

Solution. We introduce under the differential sign, thereby reducing the original integral to a tabular one.

Answer.

3. Integration by Change of Variable

Integration by change of variable, or the substitution method. Let , where the function has a continuous derivative , and there is a one-to-one correspondence between the variables and . Then the following equality holds

An integral depends on the variable of integration, so if a change of variables has been made, it is essential to return to the original variable of integration.

Integration by Change of Variable

The essence of this method is that a new variable of integration is introduced or, equivalently, a substitution is made. After that, the integral given in the problem is reduced either to a tabular integral or to one that reduces to it.

If in the indefinite integral we make the substitution , where the function is a function with a continuous first derivative, then and, by property 6 of the indefinite integral, we have:

This formula is called the formula for change of variable in an indefinite integral.

Remark

After finding the integral in the new variable , it is necessary to return to the original variable .

Remark

In some cases it is advisable to make the substitution , and then

Example

Problem. Find the integral

Solution. We replace the denominator with the variable and reduce the original integral to a tabular one.

Answer.

4. Integration by Parts Is Integration Using the Formula

When finding the function from its differential , any value of the constant of integration can be taken, since it does not appear in the final result. Therefore, for convenience, we will take .

Using the integration by parts formula is advisable in cases where differentiation simplifies one of the factors, while integration does not complicate the other.

The Method of Integration by Parts

Consider the functions and , which have continuous derivatives. By the properties of differentials, the following equality holds:

Integrating the left and right sides of the last equality, we obtain:

We rewrite the resulting equality in the form:

This formula is called the integration by parts formula. With its help, the integral can be reduced to finding the integral , which may be simpler.

Example

Problem. Find the integral

Solution. In the original integral we identify the functions and , and then perform integration by parts.

Answer.

See Also

• Derivative
• Differential
• Integral