What is the difference between a definite and indefinite integral?

A definite integral has limits of integration and the answer is a specific area. An indefinite integral returns a function of the independent variable(s).

A definite integral has limits of integration, for example:

##int_a^b f(x)dx##

where ##a## and ##b## are the limits of integration. The answer which we get is a specific area.

Most of the time, we solve the integral to get a new function of ##x## which is actually the indefinite integral and then plug in the limits, i.e.

##int_a^b f(x)dx = [g(x) +c ]_a^b =(G(b)+c)-(g(a)+c)## ##=g(b)-g(a)##

Where ##c## is an arbitrary constant, which in a real problem represents the starting conditions. You can see that it cancels out in the definite integral case anyway.

An indefinite integral has no limits specified:

##int f(x)dx = g(x) + c##

and so returns a function of the independent variable(s) plus an arbitrary constant. In this case it is easy to see that when we take the derivative of both sides, the constant disappears, which is why it is a possible part of the solution of the integral in the first place.

##d/(dx)int f(x) = f(x) = d/(dx) (g(x)+c) = d/(dx)g(x)##

ASIDE:

Sometimes the indefinite integral is not possible to find in a closed form (or without special functions that only have approximate forms), but definite integrals with specific limits are possible to find through various tricks. For example, the total area under the bell curve

##int_(-oo)^oo e^(-x^2)dx=sqrt(pi)##

Where we can't get a simple solution in terms of ##x##, but the total area is calculable! TheMathStudent on YouTube