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

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(y \right)} = y e^{x}$ and let $\operatorname{dv}{\left(y \right)} = \sin{\left(y \right)}$.

   Then $\operatorname{du}{\left(y \right)} = e^{x}$.

   To find $v{\left(y \right)}$:

   1. The integral of sine is negative cosine:

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

   Now evaluate the sub-integral.

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

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

   1. The integral of cosine is sine:

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

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

3. Now simplify:

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

4. Add the constant of integration:

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

The answer is:

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