Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ? I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting  $$i\cos(ix)+C$$
But apparently the answer is $$i\cosh(x)+C$$
I was just wondering, is this correct? And what does the "$h$" stand for/mean? Where did it even come from. Thanks in advance.

Here are two useful definitions / relations
$$\cosh(x) = \frac{e^x + e^{-x}}{2}$$
$$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$
Using these definitions you can see that
$$\cos(ix) = \frac{e^{i(ix)} + e^{-i(ix)}}{2} = \frac{e^{-x} + e^x}{2} = \cosh (x)$$
So you did get the same answer, but you just had it in a different form.
