Integration, Objectives, Integration, Integration as Inverse of Differentiation

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Objectives

After studying this lesson, you will be able to:

  • Explain integration as inverse process (anti-derivative) of differentiation;

  • Find the integral of simple functions like etc.;

  • State the following results:

  • Find the integrals of algebraic, trigonometric, inverse trigonometric and exponential functions;

  • Find the integrals of functions by substitution method.

  • Evaluate integrals of the type

  • Derive and use the result

  • State and use the method of integration by parts;

  • Evaluate integrals of the type:

  • Derive and use the result

  • Integrate rational expressions using partial fractions.

Integration

  • Integration literally means summation. Consider, the problem of finding area of region as shown in Fig.

Integration literally means summation

Integration Literally Means Summation

  • We will try to find this area by some practical method. But that may not help every time. To solve such a problem, we take the help of integration (summation) of area. For that, we divide the figure into small rectangles (See Fig.).

small rectangles (

Small Rectangles (

  • Unless these rectangles are having their width smaller than the smallest possible, we cannot find the area.

  • This is the technique which Archimedes used two thousand years ago for finding areas, volumes, etc. The names of Newton and Leibnitz are often mentioned as the creators of present day of Calculus.

  • The integral calculus is the study of integration of functions. This finds extensive applications in Geometry, Mechanics, Natural sciences and other disciplines.

  • In this lesson, we shall learn about methods of integrating polynomial, trigonometric, exponential and logarithmic and rational functions using different techniques of integration.

Integration as Inverse of Differentiation

Consider the following examples:

Let us consider the above examples in a different perspective

(i) is a function obtained by differentiation of .

is called the antiderivative of

(ii) is a function obtained by differentiation of

is called the antiderivative of

(iii) Similarly, is called the antiderivative of

Generally we express the notion of antiderivative in terms of an operation. This operation is called the operation of integration. We write

1. Integration of is

2. Integration of is

3. Integration of is

The operation of integration is denoted by the symbol

Thus

Remember that is symbol which together with symbol denotes the operation of integration.

The function to be integrated is enclosed between and .

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