Integral of sin(2*t) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 2 t$.

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

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

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

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

         1. The integral of sine is negative cosine:

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

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

      Now substitute $u$ back in:

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

   Method #2

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

      $$\int 2 \sin{\left(t \right)} \cos{\left(t \right)}\, dt = 2 \int \sin{\left(t \right)} \cos{\left(t \right)}\, dt$$

      1. There are multiple ways to do this integral.

         Method #1

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

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

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

         Method #2

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

            Then let $du = \cos{\left(t \right)} dt$ 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:

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

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

2. Add the constant of integration:

   $$- \frac{\cos{\left(2 t \right)}}{2}+ \mathrm{constant}$$

The answer is: