ANTIDERIVATIVE. Indefinite integral and its properties. Table of integrals. Direct integration, integration with the change of variables and by parts

LECTURE PLAN:

1. Antiderivative and indefinite integral

2. Main methods of integration

ANTIDERIVATIVE AND INDEFINITE INTEGRAL

Concept of antiderivative and indefinite integral

Definition 1. A function is an antiderivative of on an interval if for all in .

Theorem 1. If is an antiderivative of on an interval , then is an antiderivative of

on the interval if and only if is of the form , for all in where is a constant.

The constant is called the constant of integration.

The family of functions represented by is the general antiderivative of

is the general solution of the differential equation .

Notation for antiderivatives

When solving a differential equation of the form , we solve for , giving us the equivalent differential form . The operation of finding all solutions of this equation is called antidifferentiation or indefinite integration and is denoted by an integral sign . The general solution is denoted by

Definition. Suppose f is a function defined on an interval I and suppose further that f has an antiderivative on the interval I. The family of all antiderivatives of f is called the indefinite integral of f and is denoted by the symbol (read the integral of f with respect to x). In this notation the function is called the integrand of the indefinite integral. The process of finding the indefinite integral is called integration or, sometimes anti-differentiation. More specifically, given a function f, the expression “integrating f” means “finding the indefinite integral of f”.

There is a difference between an antiderivative and the indefinite integral. An antiderivative is a member of the family of functions represented by the indefinite integral. In the above example, , where C is a constant. The functions , and are all members of the family and are all antiderivatives of the function . It is also correct to write , where K is a constant. The difference is the choice of constants, where in this case .

Notice that all these curves are “parallel” in the sense that they never cross each other since they are translates of the function . To get a particular antiderivative one needs to know a specific point the antiderivative passes through. For example, if we want the curve that is a member of the family that passes through the point , we have and . So and it follows that .

The process of integration involves finding one antiderivative of the given function so that the indefinite integral is that one antiderivative plus the constant of integration. This can be easy for certain functions because we know the differentiation formulae. For example, because , it follows that , C a constant. Here are the integration formulae that follow directly from the appropriate differentiation formulae: The symbol C is a constant (called the constant of integration). For any differentiable function it is always the case that meaning that the function is a member of the family .

1.

2. for any constant . Notice that when , the

formula reduces to or just .

3.

4.

5.

6.

7.

8.

9.

10.

In fact every differentiation formula provides an integration formula. Consider the differentiation formula . This is a statement that two derivatives are equal. So the families of antiderivatives are also the same families. Therefore . In the same manner,

.

For reference, number these formulae as

11.

12.

These two formulae and those above can be combined to produce integrals of more complicated functions as shown in the examples below.

Properties of indefinite integral. The table of elementary integrals

1.

2.

3.

4.

5. If , then , where .

The table of indefinite integrals

Examples

Example 1: Evaluate

Solution: (Formula 12, applied twice)

(Formula 11)

(Formula 2)

(Arithmetic)

Example 2: Evaluate

Solution: (Formula 12)

(Formula 11)

(Algebra)

(Formula 2)

(Arithmetic)

Example 3: Find all the antiderivatives of .

Solution: This question is just another way to ask that be evaluated.

(Formula 12)

(Formula 11)

(Formulae 2 and 7)

(Arithmetic)

Example 4: Evaluate

Solution: (Algebra)

(Trig Relationships)

(Formula 9)

Example 5: Evaluate .

Solution: (Algebra)

(Formula 12)

(Formula 11)

(Formulae 5 and 6)

(Algebra)

Example 6: Suppose . Find the function

Solution:

(Formula 12)

(Formula 11)

(Formulae 2 and 6)

(Algebra)

(Formula 12)

(Formula 11)

(Formulae 2 and 5)

(Algebra)

Example 7: Suppose a particle is moving along a line so that the velocity at time t is given by . The displacement at time is zero. Find the displacement at any time t.