** Final answer to the problem

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** Step-by-step Solution **

** How should I solve this problem?

- Integrate by substitution
- Integrate by partial fractions
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
- Load more...

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Simplify the expression

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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

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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

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Isolate $dx$ in the previous equation

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

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

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

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

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Replace $u$ with the value that we assigned to it in the beginning: $2x$

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

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

** Final answer to the problem

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