Integral of cos(2*pi*x/3)*sin(2*pi*x/3) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \cos{\left(\frac{2 \pi x}{3} \right)}$.

      Then let $du = - \frac{2 \pi \sin{\left(\frac{2 \pi x}{3} \right)} dx}{3}$ and substitute $- \frac{3 du}{2 \pi}$:

      $$\int \frac{9 u}{4 \pi^{2}}\, du$$

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

         $$\int \left(- \frac{3 u}{2 \pi}\right)\, du = - \frac{3 \int u\, du}{2 \pi}$$

         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: $- \frac{3 u^{2}}{4 \pi}$

      Now substitute $u$ back in:

      $$- \frac{3 \cos^{2}{\left(\frac{2 \pi x}{3} \right)}}{4 \pi}$$

   Method #2

   1. Let $u = \frac{2 \pi x}{3}$.

      Then let $du = \frac{2 \pi dx}{3}$ and substitute $\frac{3 du}{2 \pi}$:

      $$\int \frac{9 \sin{\left(u \right)} \cos{\left(u \right)}}{4 \pi^{2}}\, du$$

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

         $$\int \frac{3 \sin{\left(u \right)} \cos{\left(u \right)}}{2 \pi}\, du = \frac{3 \int \sin{\left(u \right)} \cos{\left(u \right)}\, du}{2 \pi}$$

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

            Then let $du = - \sin{\left(u \right)} du$ 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:

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

         So, the result is: $- \frac{3 \cos^{2}{\left(u \right)}}{4 \pi}$

      Now substitute $u$ back in:

      $$- \frac{3 \cos^{2}{\left(\frac{2 \pi x}{3} \right)}}{4 \pi}$$

   Method #3

   1. Let $u = \sin{\left(\frac{2 \pi x}{3} \right)}$.

      Then let $du = \frac{2 \pi \cos{\left(\frac{2 \pi x}{3} \right)} dx}{3}$ and substitute $\frac{3 du}{2 \pi}$:

      $$\int \frac{9 u}{4 \pi^{2}}\, du$$

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

         $$\int \frac{3 u}{2 \pi}\, du = \frac{3 \int u\, du}{2 \pi}$$

         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: $\frac{3 u^{2}}{4 \pi}$

      Now substitute $u$ back in:

      $$\frac{3 \sin^{2}{\left(\frac{2 \pi x}{3} \right)}}{4 \pi}$$

2. Now simplify:

   $$- \frac{3 \cos^{2}{\left(\frac{2 \pi x}{3} \right)}}{4 \pi}$$

3. Add the constant of integration:

   $$- \frac{3 \cos^{2}{\left(\frac{2 \pi x}{3} \right)}}{4 \pi}+ \mathrm{constant}$$

The answer is:

$$- \frac{3 \cos^{2}{\left(\frac{2 \pi x}{3} \right)}}{4 \pi}+ \mathrm{constant}$$