Integral of sin(x/2)cos(x/2) dx

1. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int 2 u\, du$$

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

         $$\int u\, du = 2 \int u\, du$$

         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: $u^{2}$

      Now substitute $u$ back in:

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

   Method #2

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

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

      $$\int 2 \sin{\left(u \right)} \cos{\left(u \right)}\, du$$

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

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

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

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

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

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

               $$\int u\, du = - \int u\, du$$

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

      Now substitute $u$ back in:

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

   Method #3

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

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

      $$\int \left(- 2 u\right)\, du$$

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

         $$\int u\, du = - 2 \int u\, du$$

         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: $- u^{2}$

      Now substitute $u$ back in:

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

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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