Question

Why we add c while doing integration .can u explain me with the help of an example

Answer 1

Integration is an inverse function of derivation .

One way to look at it is that an indefinite integral ∫f∫f asks for a solution to the differential equation F′(x)=f(x)F′(x)=f(x). That is, the function f(x)f(x) is given and you are looking for a function F(x)F(x) such that F′(x)=f(x)F′(x)=f(x). Now, it may be that a solution does not exist but if a solution does exist, say G(x)G(x) is found such that G′(X)=f(x)G′(X)=f(x) then for any constant CC, the function G(x)+cG(x)+c is also a solution (just compute the derivative to see that).

So the existence of a single solution implies the existence of infinitely many solutions. There is no particular reason to prefer one over the other so we indicate the entire family of solution by the (magical) '+C'.

It should be noted that any two solutions of F′(x)=f(x)F′(x)=f(x) differ by a constant (to prove that consider the difference between two such solutions, and take the derivative) so that writing the (family of) solutions as F(x)+CF(x)+C very precisely gives all of the solutions.

or in simple word
derivative of (x+2)= 1
but integrel(1)=x
here you can see that 2 is missing in the answer of integration To compensate this wee add a contant c to the answer. Thus the addition of constant