Integral of 2*x*y-7cos(y)dy dx

1. Integrate term-by-term:

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

      $$\int 2 x y\, dy = 2 x \int y\, dy$$

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

         $$\int y\, dy = \frac{y^{2}}{2}$$

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

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

      $$\int \left(- 1 \cdot 7 \cos{\left(y \right)}\right)\, dy = - \int 7 \cos{\left(y \right)}\, dy$$

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

         $$\int 7 \cos{\left(y \right)}\, dy = 7 \int \cos{\left(y \right)}\, dy$$

         1. The integral of cosine is sine:

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

         So, the result is: $7 \sin{\left(y \right)}$

      So, the result is: $- 7 \sin{\left(y \right)}$

   The result is: $x y^{2} - 7 \sin{\left(y \right)}$

2. Add the constant of integration:

   $$x y^{2} - 7 \sin{\left(y \right)}+ \mathrm{constant}$$

The answer is:

$$x y^{2} - 7 \sin{\left(y \right)}+ \mathrm{constant}$$