Integral of cos(x)×e^sin(x) dx

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

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

   $$\int 1\, du$$

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

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

   Now substitute $u$ back in:

   $$e^{\sin{\left(x \right)}}$$

2. Now simplify:

   $$e^{\sin{\left(x \right)}}$$

3. Add the constant of integration:

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

The answer is:

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