If f(x) is a function, then an antiderivative of f(x) is another function F(x), such that F'(x)=f(x).
Intergral:
If f(x) is a function, then the integral of f(x) over the interval [a,b] is the limit of its Riemann Sum over that interval (if it converges).
Difference
For anti differentiation , if there is an anti derivative, there are infinite function F(x): F(x)+1, F(x)+1/2, F(x)+pi, they can be concluded by F(x)+r,r is any real number. There is no special antiderivative for a function, they’re all on equal footing.
However, integral gives the area under the function f(x) between x=a and x=b.
This means that the integral of a function is a number that is special.
Take an example, we have a function f(x)=3x^2.
The anti derivative can be F(x)=x^3+1,or F(x)=x^3+3…
However,the integral from 1 to 2 is \int_1^2 3x^2 dx=8, which is special.
So we know that anti derivative is not special for a function but integral is special for a function.
