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
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 finding the derivative of 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
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: $2x$
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 $C$
Try other ways to solve this exercise
Get a preview of step-by-step solutions.
Earn solution credits, which you can redeem for complete step-by-step solutions.
Save your favorite problems.
Become premium to access unlimited solutions, download solutions, discounts and more!
