Indefinite Integration

Integration is the inverse process of differentiation. If the differentiation of a function g(x) is f(x), then the integration of the function f(x) with respect to x is g(x) plus a constant of integration.
From the geometric point of view, an indefinite integral is a collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.
Integral calculus is used due to the efforts of solving the problems of the following types:
(a) the problem of finding a function if its derivative is given,
(b) the problem of finding the area bounded by the graph of a function under certain conditions.

Properties of indefinite integrals:
1.    The derivative of an integral is the integrand itself.
2.    Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.
3.    The integral of the sum or difference of a finite number of functions is equal to the sum or difference of the integrals of the individual functions.
4.    The constant can be taken outside the integral sign.

Comparison between differentiation and integration:
1.    Both are operation on functions.
2.    Both satisfy the property of linearity.
3.    Both operations are not always applicable to all functions. Some functions are not differentiable. Similarly, all functions are not integrable.
4.    Derivative of a function, if exists, is unique. The integral of a function represents a family of the curves.
5.    When a polynomial function P is differentiated, the degree of its derivative decreases by 1. When it is integrated, the degree of the integral increases by 1.

Keywords: Standard formulae for integration, Geometrical interpretation of Indefinite Integrals

 

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