CHAPTER 6 : INTEGRATION
We know that in algebra the operations of addition and subtraction, multiplication and division form reverse pairs of operations. In calculus the operations of derivative and integration (antiderivative) are also thought of an inverse of one another. Thus they also form a reverse pair of operations and the rules for finding derivatives will be useful in establishing corresponding rules for finding antiderivatives.
6.1 Anti  Derivatives ( Indefinite Integrals )
If a function f exists , for a given function f, such that f˘ ( x ) = f ( x ), then f is called antiderivative of f.
An antiderivative is also known as primitive or an indefinite integral.
Symbolically, this is written as and is read as " f ( x ) is an integral of f ( x )" w.r. to x .
Here f ( x ) is called the integrand and the process of finding the integral is called integration.
The above definition poses the following questions.
(1) Does every function poses an antiderivative?
(2) Is an antiderivative unique in case it exists?
(3) Is it always possible to find an antiderivative?
The following answers are suggested :
(1) There exist a large number of functions possessing an
antiderivative, e.g. all continuous functions. In finding
an antiderivative, we shall always assume its existence. However
every function need not posses an antiderivative.
(2) Let so that
for any constant ‘c’
so that f ( x ) + c is also an antiderivative of f ( x ). This accounts for the name "Indefinite Integral".
(3) In the present chapter, we shall study several methods of finding antiderivatives. This, however, does not ensure that antiderivative of any given function can always be obtained.
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