Question

Can someone give an example of an arbitrary constant?

Answer 1

Do you know how to take the derivative of
f(x) = x ?

you get
f'(x) = 1

however, if you had
f(x)= x + 3
and take the derivative of this function you get
f'(x) = 1 + d/dx(3)
f'(x) = 1

in other words, the same answer as for the first equation.

In fact, if we have
f(x) = x + *any constant*
we find
f'(x) = 1

When we "anti differentiate" or "integrate", we do not get back the original equation (and cannot , unless we are given more information). To show this, we write
\[ \int f'(x) = f(x) + C \]
where C stands for an unknown constant which the original equation has (though C could be 0)

Answer 2

If your question is asking something else, please post again.

Answer 3

Thanks, that was pretty helpful, but I was just wondering how can I give a simple example to just portray the arbitrarily constant at work. Thanks.