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

1. There are multiple ways to do this integral.

   Method #1

   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:

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

   Method #2

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

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

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

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

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

         1. The integral of the exponential function is itself.

            $$\int e^{u}\, du = e^{u}$$

         So, the result is: $- e^{u}$

      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:

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