Integral of (e^(x+sin(y)))*cos(y) dy

1. There are multiple ways to do this integral.

   Method #1

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

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

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

      1. The integral of the exponential function is itself.

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

      Now substitute $u$ back in:

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

   Method #2

   1. Rewrite the integrand:

      $$e^{x + \sin{\left(y \right)}} \cos{\left(y \right)} = e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}$$

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

      $$\int e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy = e^{x} \int e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy$$

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

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

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

         1. The integral of the exponential function is itself.

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

         Now substitute $u$ back in:

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

      So, the result is: $e^{x} e^{\sin{\left(y \right)}}$

   Method #3

   1. Rewrite the integrand:

      $$e^{x + \sin{\left(y \right)}} \cos{\left(y \right)} = e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}$$

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

      $$\int e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy = e^{x} \int e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy$$

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

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

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

         1. The integral of the exponential function is itself.

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

         Now substitute $u$ back in:

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

      So, the result is: $e^{x} e^{\sin{\left(y \right)}}$

2. Add the constant of integration:

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

The answer is:

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