Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Choose an option
- Integrate by partial fractions
- Integrate by substitution
- 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
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Simplify $2\sin\left(x\right)\cos\left(x\right)$ using the trigonometric identity: $\sin(2x)=2\sin(x)\cos(x)$
Simplify the fraction $\frac{2\sin\left(2x\right)}{2}$ by $2$
Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$
Simplify $\frac{1}{2\sin\left(x\right)\cos\left(x\right)}$ into $\csc\left(2x\right)$ by applying trigonometric identities
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
Rearrange the equation
Divide both sides of the equation by $2$
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
The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$
Multiply the fraction and term in $-\left(\frac{1}{2}\right)\ln\left|\csc\left(u\right)+\cot\left(u\right)\right|$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
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 $C$
