Integrating $e^x\sinh x$ gives two different answers

Question

I integrated $e^x\sinh(x)$ by writing out $\sinh x$ as $\frac{e^x-e^{-x}}{2}$ and multiplying to get

$\int$$\frac{e^{2x}-e^{0}}{2}dx$

which comes out to be

$\frac{e^{2x}}{4}- \frac{x}{2}$ + C

however, plugging the question into photo math they use integration by parts and the answer is instead:

$\frac{e^{2x}-2}{4}- \frac{x}{2}$ + C

I was wondering which answer is correct and why does solving it in different ways give different answers?