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Concept of Indefinite Integral
In differential calculus, we have learned how to calculate the differential coefficient f ′ (x) of a given function f (x) with respect to x. In this chapter, we have to find out the primitive function f (x) (i.e. original function) whenever its derived function f ′ (x) (i.e. derivative of a function) is given, such process is called integration or anti differentiation.
∴ Integration is the reverse process of differentiation
We know that d/dx (sin x ) = cos x. Here cos x is known as Derived function, and sin x is known as Primitive function [also called as Anti derivative function (or) Integral function].
A function F (x) is said to be a primitive function of the derived function f (x) , if d/dx [ F (x)]= f (x)
Now, consider the following examples which are already known to us.
From the above examples, we observe that 3x2 is the derived function of the primitive functions x3 , x3 + 5 , x3 – 3/2 , x 3 + e , x3 − π , ... and which indicates that the primitive functions are need not be unique, even though the derived function is unique. So we come to a conclusion that x 3 + c is the primitive function of the derived function 3x2 .
∴ For every derived function, there are infinitely many primitives by choosing c arbitrarily from the set of real numbers R. So we called these integrals as indefinite integrals.
The process of determining an integral of a given function is defined as integration of a function.
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