Question

What is the difference between an antiderivative and an integral?

Answer 1

Hint : When we talk in general, we frequently say that integral and derivative are the exactly two opposite processes. But here we will exactly get the difference between the two. With the help of an example, we will clarify the concept.

Complete step by step solution:
We know that finding the integral of any function sometimes is treated as finding the antiderivative of the same function. But these two are different processes. How? Let’s get the clarification.
Integration is of two different types. One is a definite integral and the other is an indefinite integral.
Now what are these two things?
Definite integral is having upper and lower limits for the function that is getting integrated.
Whereas in indefinite integral there are no such limits.
When an indefinite integral is found there is some constant C. That makes the function totally different from the antiderivative if taken.
But in the case of definite integrals there are limiting terms that give the perfect function in return. That is there is no big difference between a definite integral and an antiderivative. Rather there is a big difference in antiderivative and indefinite integral.
For example,
$f\left( x \right) = {x^2}$
This is to be integrated or we can say generally to be antiderivative.
Now the definite integral that has the upper and lower limits will give the answer as $\dfrac{1}{3}\left( {{b^3} - {a^3}} \right)$ where a and b are the lower and upper limits respectively.
Now the indefinite integral will be $\dfrac{1}{3}{x^3} + C$ this constant C will only differ it from the antiderivative.

Note : Note that we are just differing the antiderivative and integral with the help of two types of integrals only. The example so used is just to clarify the two separate things. But in general language we use this as integral or antiderivative.