Integral of -xsin(2x)+2xcos(y)sin(y) dy

1. Integrate term-by-term:

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

      $$\int - x \sin{\left(2 x \right)}\, dy = - x y \sin{\left(2 x \right)}$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Let $u = \cos{\left(y \right)}$.

         Then let $du = - \sin{\left(y \right)} dy$ and substitute $- 2 du x$:

         $$\int \left(- 2 u x\right)\, du$$

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

            $$\int u\, du = - 2 x \int u\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u\, du = \frac{u^{2}}{2}$$

            So, the result is: $- u^{2} x$

         Now substitute $u$ back in:

         $$- x \cos^{2}{\left(y \right)}$$

      Method #2

      1. Let $u = \sin{\left(y \right)}$.

         Then let $du = \cos{\left(y \right)} dy$ and substitute $2 du x$:

         $$\int 2 u x\, du$$

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

            $$\int u\, du = 2 x \int u\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u\, du = \frac{u^{2}}{2}$$

            So, the result is: $u^{2} x$

         Now substitute $u$ back in:

         $$x \sin^{2}{\left(y \right)}$$

   The result is: $- x y \sin{\left(2 x \right)} - x \cos^{2}{\left(y \right)}$

2. Now simplify:

   $$- x \left(y \sin{\left(2 x \right)} + \cos^{2}{\left(y \right)}\right)$$

3. Add the constant of integration:

   $$- x \left(y \sin{\left(2 x \right)} + \cos^{2}{\left(y \right)}\right)+ \mathrm{constant}$$

The answer is:

$$- x \left(y \sin{\left(2 x \right)} + \cos^{2}{\left(y \right)}\right)+ \mathrm{constant}$$