Integral of ecos(x)*sin(x)*dx dx

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

   $$\int e \cos{\left(x \right)} \sin{\left(x \right)} 1\, dx = e \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

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

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

         $$\int u\, du$$

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

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

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

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

            So, the result is: $- \frac{u^{2}}{2}$

         Now substitute $u$ back in:

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

      Method #2

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

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

         $$\int u\, du$$

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

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

         Now substitute $u$ back in:

         $$\frac{\sin^{2}{\left(x \right)}}{2}$$

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

2. Add the constant of integration:

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

The answer is:

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