Finding all holomorphic function such $\text{Im} f(x+ i y) = e^x(x \sin y + \sin y+ y \cos y)$
The only way for you to have $(\forall x,y\in\Bbb R):\alpha(x)=\beta(y)$ is that both $\alpha$ and $\beta$ are constant.
And if $u(x,y)=e^x(x\cos(y)+\cos(y)-y\sin(y))+1$ and $v(x,y)=e^x(x\sin(y)+\sin(y)+y\cos(y))$, then\begin{align}f(x+yi)&=u(x,y)+v(x,y)i\\&=e^x(x\cos(y)+\cos(y)-y\sin(y))+1+\bigl(e^x(x\sin(y)+\sin(y)+y\cos(y))\bigr)i\\&=e^x\bigl(x\cos(y)+\cos(y)-y\sin(y)+(x\sin(y)+\sin(y)+y\cos(y))i\bigr)+1\\&=e^x(\cos(y)+\sin(y)i)(x+1+yi)+1\\&=e^{x+yi}\bigl((x+yi)+1\bigr)+1.\end{align}