The standard definition of $\int_a^bf(x)dx$ is the difference of the antiderivatives at each of the bounds. Simple enough.
But if the notation for the antiderivative is $\int f(x)dx$… how do you indicate the value of that antiderivative at a particular x-value? What is the math notation for "the antiderivative of $f$ at $x=a$"?
(I used to write it simply as $\int_0^af(x)dx$, but for functions like $\frac{1}{x}$, that lower bound may not be defined.)
Salaam! The integral in question is indefinite integral, which means we do not know the lower or upper limits. In this case, we can still calculate the integral (area under the curve) as it is isomorphic up to a constant as the derivative of a constant is zero.
For example:
Let $$f(x)=2x^2 +5$$ $$f'(x)=4x+0=4x$$ Hence, by the fundamental theorem of calculus, we have:
$$f(x) = \int f'(x) dx= \int 4x dx = 2x^2 + constant$$ In this particular case, we know that the constant is 5.
Now if you want to compute $\int f'(x) dx$ at $x =5$, it is $f(5)=2(5)^2 +5$
I hope this helps, and please let me know if you still have any questions!