Integral of sinxe^cosx dx

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

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

   $$\int 1\, du$$

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

      $$\int \left(-1\right)\, du = - \int 1\, du$$

      1. The integral of a constant is the constant times the variable of integration:

         $$\int 1\, du = u$$

      So, the result is: $- u$

   Now substitute $u$ back in:

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

2. Add the constant of integration:

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

The answer is:

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