We can solve the integral $\int\csc\left(2x\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Substituting $u$ and $dx$ in the integral and simplify

Take the constant $\frac{1}{2}$ out of the integral

The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$

Multiply the fraction and term in $-\left(\frac{1}{2}\right)\ln\left|\csc\left(u\right)+\cot\left(u\right)\right|$

Simplify $\csc\left(2x\right)+\cot\left(2x\right)$ using trigonometric identities

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$