The integration constant can be any constant. For a definite integral it is determined by the boundary condition or initial value. But if g(x) is an antiderivative of f(x), then so is g(x) + c, with c being independent of x, since then dc(x)/dx = 0. And your quite welcome, Guillermo. Calculus is a lot of fun.

Take log(sin(x)) + (cos^2(x))/2 and substitute cos^2(x) = 1-sin^2(x) and you get log(sin(x)) - (sin^2(x))/2 + 1/2

The 1/2 disappears when you differentiate, so both are correct. Another way to look at it is the indefinite integral comes with an integration constant, which can consume the 1/2.