Integral of cos(2*x)*sin(x) dx

1. Rewrite the integrand:

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

2. Integrate term-by-term:

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

      $$\int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

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

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

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

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

            $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

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

               $$\int u^{2}\, du = \frac{u^{3}}{3}$$

            So, the result is: $- \frac{u^{3}}{3}$

         Now substitute $u$ back in:

         $$- \frac{\cos^{3}{\left(x \right)}}{3}$$

      So, the result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3}$

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

      $$\int \left(- \sin{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)}\, dx$$

      1. The integral of sine is negative cosine:

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

      So, the result is: $\cos{\left(x \right)}$

   The result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}$

3. Now simplify:

   $$\frac{\left(2 - \cos{\left(2 x \right)}\right) \cos{\left(x \right)}}{3}$$

4. Add the constant of integration:

   $$\frac{\left(2 - \cos{\left(2 x \right)}\right) \cos{\left(x \right)}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(2 - \cos{\left(2 x \right)}\right) \cos{\left(x \right)}}{3}+ \mathrm{constant}$$