Integral of e^cos2x*sin2xdx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \cos{\left(2 x \right)}$.

      Then let $du = - 2 \sin{\left(2 x \right)} dx$ and substitute $- \frac{du}{2}$:

      $$\int \left(- \frac{e^{u}}{2}\right)\, du$$

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

         $$\text{False}$$

         1. The integral of the exponential function is itself.

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

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

      Now substitute $u$ back in:

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

   Method #2

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

      $$\int 2 e^{\cos{\left(2 x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int e^{\cos{\left(2 x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\, dx$$

      1. Rewrite the integrand:

         $$e^{\cos{\left(2 x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} = e^{2 \cos^{2}{\left(x \right)} - 1} \sin{\left(x \right)} \cos{\left(x \right)}$$

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

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

         $$\int \left(- \frac{u e^{2 u^{2}}}{e}\right)\, du$$

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

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

            1. Let $u = 2 u^{2}$.

               Then let $du = 4 u du$ and substitute $\frac{du}{4}$:

               $$\int \frac{e^{u}}{4}\, du$$

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

                  $$\text{False}$$

                  1. The integral of the exponential function is itself.

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

                  So, the result is: $\frac{e^{u}}{4}$

               Now substitute $u$ back in:

               $$\frac{e^{2 u^{2}}}{4}$$

            So, the result is: $- \frac{e^{2 u^{2}}}{4 e}$

         Now substitute $u$ back in:

         $$- \frac{e^{2 \cos^{2}{\left(x \right)}}}{4 e}$$

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

2. Add the constant of integration:

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

The answer is:

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