Integral of 12*ctg(3*x) dx

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

   $$\int 12 \cot{\left(3 x \right)}\, dx = 12 \int \cot{\left(3 x \right)}\, dx$$

   1. Rewrite the integrand:

      $$\cot{\left(3 x \right)} = \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$$

   2. There are multiple ways to do this integral.

      Method #1

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

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

         $$\int \frac{1}{3 u}\, du$$

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

            $$\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$$

            1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

            So, the result is: $\frac{\log{\left(u \right)}}{3}$

         Now substitute $u$ back in:

         $$\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{3}$$

      Method #2

      1. Let $u = 3 x$.

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

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

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

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

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

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

               $$\int \frac{1}{u}\, du$$

               1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

               Now substitute $u$ back in:

               $$\log{\left(\sin{\left(u \right)} \right)}$$

            So, the result is: $\frac{\log{\left(\sin{\left(u \right)} \right)}}{3}$

         Now substitute $u$ back in:

         $$\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{3}$$

   So, the result is: $4 \log{\left(\sin{\left(3 x \right)} \right)}$

2. Add the constant of integration:

   $$4 \log{\left(\sin{\left(3 x \right)} \right)}+ \mathrm{constant}$$

The answer is:

$$4 \log{\left(\sin{\left(3 x \right)} \right)}+ \mathrm{constant}$$