Integral of xsin(x+y) dx

1. Use integration by parts:

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

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

   Then $\operatorname{du}{\left(x \right)} = 1$.

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

   1. Let $u = x + y$.

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

      $$\int \sin{\left(u \right)}\, du$$

      1. The integral of sine is negative cosine:

         $$\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$$

      Now substitute $u$ back in:

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

   Now evaluate the sub-integral.

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

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

   1. Let $u = x + y$.

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

      $$\int \cos{\left(u \right)}\, du$$

      1. The integral of cosine is sine:

         $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

      Now substitute $u$ back in:

      $$\sin{\left(x + y \right)}$$

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

3. Add the constant of integration:

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

The answer is:

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