Why does the constant of integration move?

Question

I'm trying to make sense of a solution to a separation of variables question. Namely where it goes from:

$$y = \exp(-\cos x + C)$$

To:

$$y = A\cdot \exp(-\cos x)$$

I understand the constant of integration can be quite flexible (oxymoron) in its undefined state but I'm trying to track down an explanation for this particular "identity". This isn't one of the standard exponential identities.

Answer 1

This is because of the addition exponent rule in disguise, with base $b$ and exponents $d$ and $c$:

$$b^{d+c} = b^d \cdot b^c$$

In this case:

$$\exp(-\cos x + C) = e^{-\cos x + C} = e^{-\cos x}\cdot e^C$$

And by renaming $e^C$ to $A$, since both are just constants, we get:

$$A \cdot e^{-\cos x} = A\cdot\exp(-\cos x)$$

Answer 2

We get $$e^{C-\cos(x)}=e^C\cdot e^{-\cos(x)}$$

Answer 3

One has $$ e^{-\cos x+C}=e^{C}\cdot e^{-\cos x}=A\cdot e^{-\cos x} $$ with $$ A=e^{C}. $$