Integral of exp(y-x)*sin(y) dx

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

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

   1. Let $u = - x + y$.

      Then let $du = - dx$ and substitute $- du$:

      $$\int \left(- e^{u}\right)\, du$$

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

         $$\text{False}$$

         1. The integral of the exponential function is itself.

            $$\int e^{u}\, du = e^{u}$$

         So, the result is: $- e^{u}$

      Now substitute $u$ back in:

      $$- e^{- x + y}$$

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

2. Add the constant of integration:

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

The answer is:

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