Integral of e^x*sin(y)-1 dy

1. Integrate term-by-term:

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int e^{x} \sin{\left(y \right)}\, dy = e^{x} \int \sin{\left(y \right)}\, dy$$

      1. The integral of sine is negative cosine:

         $$\int \sin{\left(y \right)}\, dy = - \cos{\left(y \right)}$$

      So, the result is: $- e^{x} \cos{\left(y \right)}$

   1. The integral of a constant is the constant times the variable of integration:

      $$\int \left(-1\right)\, dy = - y$$

   The result is: $- y - e^{x} \cos{\left(y \right)}$

2. Add the constant of integration:

   $$- y - e^{x} \cos{\left(y \right)}+ \mathrm{constant}$$

The answer is:

$$- y - e^{x} \cos{\left(y \right)}+ \mathrm{constant}$$