Are both of these indefinite integrals?

Solution 1:

Not quite. (1) is definitely (no pun intended) an indefinite integral. Note that evaluating an indefinite integral does not actually lead to a function, but rather to a family of functions differing by a constant. In the case of (1), we have $f(x) = \sin x + C$ (a family of functions).

(2) is simply a function for any given $a$: by the Fundamental Theorem of Calculus, $g(x) = \sin x - \sin a$. Note that this really is one function, not a family of functions. (If one takes a family of parameters $a$, then that would result in a family of functions $g(x) = g_a(x)$; but it still wouldn't result in every possible function in the family from (1), since $\sin a$ is bounded.)