Final Answer
Step-by-step Solution
Problem to solve:
Specify the solving method
Simplify $2\sin\left(x\right)\cos\left(x\right)$ using the trig identity: $\sin(2x)=2\sin(x)\cos(x)$
Simplify the fraction $\frac{1}{1\sin\left(2x\right)}$ by $1$
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Simplify the expression inside the integral
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
Differentiate both sides of the equation $u=2x$
Find the derivative
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Divide $1$ by $2$
The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
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 $C$