Integral of sinx*cos(cosx) dx

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

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

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

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

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

      1. The integral of cosine is sine:

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

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

   Now substitute $u$ back in:

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

2. Add the constant of integration:

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

The answer is:

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