Integral of xsin(5x) 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 $\operatorname{dv}{\left(x \right)} = \sin{\left(5 x \right)}$.

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

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

   1. Let $u = 5 x$.

      Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{\sin{\left(u \right)}}{5}\, du$$

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

         $$\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$$

         1. The integral of sine is negative cosine:

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

         So, the result is: $- \frac{\cos{\left(u \right)}}{5}$

      Now substitute $u$ back in:

      $$- \frac{\cos{\left(5 x \right)}}{5}$$

   Now evaluate the sub-integral.

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

   $$\int \left(- \frac{\cos{\left(5 x \right)}}{5}\right)\, dx = - \frac{\int \cos{\left(5 x \right)}\, dx}{5}$$

   1. Let $u = 5 x$.

      Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{\cos{\left(u \right)}}{5}\, du$$

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

         $$\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$$

         1. The integral of cosine is sine:

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

         So, the result is: $\frac{\sin{\left(u \right)}}{5}$

      Now substitute $u$ back in:

      $$\frac{\sin{\left(5 x \right)}}{5}$$

   So, the result is: $- \frac{\sin{\left(5 x \right)}}{25}$

3. Add the constant of integration:

   $$- \frac{x \cos{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{25}+ \mathrm{constant}$$

The answer is: