# Integration Rules and Integration definition with examples

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## A Complete Preparation Book for Integration (Calculus)

Integration is very important part of Calculus,
Integration is the reverse of Differentiation.
Basic examples of Integration rules
If     y = 2x + 7
or    y = 2x – 8
or    y = 2x + 100000

then for all cases  dy/dx = 2
Hence we can say that anti-derivative or
integral of 2 is 2x + C
where C = any real constant.
Can you guess what is the integration of 2x ? well to learn Integration completely you can download my free eBook named Indefinite and Definite Integration Rules and Theory with Examples and Exercise

Q: What are the Integrate Synonym?

Ans: Integrate Synonym or another word for Integrate are Integral or Integration or primitive or an anti-derivative of a function.

## Integration Definition

A function f(x) is called an integration or Integral or primitive or anti-derivative of a function g(x) if f'(x) = g(x)
The process of finding an indefinite integration of a given function is called integral of the function.

## Basic Integration Rules using integration definition

(1) The differentiation of an integral is the integrand itself or the process of differentiation and integral neutralize each other.
(2) The integral of the product of a constant and a function is equal to the product of the constant and the integral of the function.
(3) Integral of the sum or difference of two functions is equal to the sum or difference of their integrals.
(4) The signum function has an antiderivative on any interval which doesn’t contain the point x = 0, and does not possess an anti-derivative on any interval which contains the point.
(5) The antiderivative of every odd function is an even function and vice-versa

## Comparison between Differentiation and Integration.

(1) Differentiation and integration both are operations on functions and each gives rise to a function.
(2) we can not do differentiation and integration of all functions.
(3) The derivative of a function, if it exists, is unique. The integration of a function, if it exists, is not unique.
(4) The derivative of a polynomial function decreases its degree by 1, but the integration of a polynomial function increases its degree by 1.
(5) The derivative has a geometrical meaning, namely, the slope of the tangent to a curve at a point on it. The integration has also a geometrical meaning, namely, the area of some region.
(6) The derivative is used in obtaining some physical quantities like velocity, acceleration etc. of a particle. The integration is used in obtaining some physical quantities like centre of mass, momentum etc.
(7) Differentiation and integration are inverse of each other.

## CONTENTS OF INTEGRATION RULES AND DEFINITION WITH EXAMPLES COMPLETE THEORY AND EXERCISE.

Integration definition and Integrated meaning

Comparison between Differentiation and Integration

Integration Rules

Properties of Integration

Methods to find Integral

Fundamental Integration Rules

Integration Rules by Substitution

Integration Rules by Parts formula

Integral Calculus formulas

Evaluation of the various forms of Integrals by use of Standard results

Integration Rules of different standard forms

Integration Rules of rational functions by using Partial fractions

Integration Rules of Trigonometric functions

Integration Rules using Euler’s substitution

Some integrals which can not be found.

Definite Integral as the Limit of a Sum Rules

Definite Integration by Substitution Rules

Properties of Definite Integration

Summation of Series by Integration Rules

Solving Integral Equations

Derivatives of Integrals Rules

Gamma Function Rules

Reduction formula for Definite Integration Rules

Walli’s Formula for Integration Rules

Leibnitz’s Integration Rules

Integration Rules with Infinite Limits(Improper Integrals)

Some important Definite Integration Rules

Integration Rules of Piece-wise Continuous Function

Integral Calculus Examples

## HISTORY OF INTEGRATION

The origin of the integration goes back to the early period of development of mathematics and it is related to the method of exhaustion developed by the mathematicians of ancient Greece. This method arose in the solution of problems on calculating areas and volumes of solid bodies etc. In this sense, the method of exhaustion can be regarded as an early method of Integration.

The greatest development in integration is method of exhaustion in the early period was obtained in the works of Eudoxus (440 B.C.) and Archimedes (300 B.C.) Conclusively, the fundamental concepts and theory and integration and primarily its relationship with differential calculus were developed in the work of P.de Fermat, I. Newton and G.Leibnitz at the end of 17th century A.D.

Newton introduced the basic notion of inverse function called the anti-derivative (Integration) or the inverse method of tangents.

During 1684-86 A.D. Leibnitz published an article in the Acta Eruditorum which he called Calculas summatorius, since it was connected with the summation of a number of infinitely small areas, whose sum, he indicated by the symbol ∫. In 1696 A.D. he followed a suggestion made by J.Bernoulli and changed this article to calculus integrali. This corresponded to Newton’s inverse method of tangents.

Both Newton and Leibnitz adopted quite independent lines of approach which were radically different. However, respective theories accomplished results that were practically identical. Leibnitz used the notion of definite integration and what is quite certain is that he first clearly appreciated tie up between the anti-derivative and the definite integral. The discovery that differentiation and integration are inverse operations belongs to Newton and Leibnitz.

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