Indefinite Integrals and Antiderivatives

By the Fundamental Theorem of Calculus I, since
$$\qquad F'(x)=\frac{d}{dx}\int_a^x f(t)\, dt=f(x), \qquad\text{ we have }\qquad \int f(x)\, dx = F(x) = \int_a^x f(t)\, dt.$$

This shows that the integral function is indeed an antiderivative; it is the antiderivative of its integrand.  All antiderivatives are the same, up to adding a constant, and indeed changing the value of $a$ in $F(x)=\int_a^x f(t)\, dt$ changes $F(x)$ by a constant, so both antiderivatives and integral functions are only defined up to a constant.

Don't get too bogged down in the previous two paragraphs, but you do need to understand the three types of "integrals" listed above.  From now on, the notation $\int f(x)\, dx$ will refer to the antiderivative, and we'll usually just call it "the antiderivative of $f(x)$" or "the integral of $f(x)$" (with respect to $x$) for short.