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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 Theorem 1. If The constant The family of functions represented by
Notation for antiderivatives When solving a differential equation of the form 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 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, Notice that all these curves are “parallel” in the sense that they never cross each other since they are translates of the function 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 1. 2. formula reduces to 3. 4. 5. 6. 7. 8. 9. 10. In fact every differentiation formula provides an integration formula. Consider the differentiation formula
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 The table of indefinite integrals
Examples Example 1: Evaluate Solution:
Example 2: Evaluate Solution:
Example 3: Find all the antiderivatives of Solution: This question is just another way to ask that
Example 4: Evaluate Solution:
Example 5: Evaluate Solution:
Example 6: Suppose Solution:
Example 7: Suppose a particle is moving along a line so that the velocity at time t is given by Solution:
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