Integrate Cot^n (x). Hello friends, today I’ll talk about how to integrate Cot^n (x). Have a look!!
Want to know more about the reduction formulas of other functions?? Have a look at the followings:
Reduction formulas for trigonometric functions
- Reduction formula for Sin^(n) x | Integrate Sin^(n) x dx
- Get the reduction formula for Cos^(n) x | Integrate Cos^(n) x dx
- Reduction formula for Tan^(n) x | Integrate Tan^(n) x dx
- Reduction formula for Sec^(n) x | Integrate Sec^(n) x dx
Reduction formulas for logarithmic functions
Reduction formulas for hyperbolic functions
Integrate Cot^n (x)
Solved example of how to integrate Cot^n (x)
Disclaimer: This example is not mine. I have chosen it from some book. I have also given the due reference at the end of the post.
So here is my example.
According to Stroud and Booth (2013)*, “If , show that
Hence determine .”
Now here I have to integrate , that is,
Also I have to prove that
First of all, I’ll write as
Also, from the standard formulas in trigonometry, I know that
Therefore will be
So will be
Then I’ll simplify it as
And that gives
Also, if , then
Now I’ll integrate .
Since I already know that , I can rewrite as
Thus will be
If , then . Thus will be
Now I’ll integrate the right-hand side to get
Next, I’ll simplify it. And that gives
Hence I have proved the relation. And this is the first part of the given example.
Next, I’ll get the value of .
So I’ll substitute in equation (3) to get
Now I need the value of to get the value of . So I’ll substitute in equation (3) to get
Again I have to get the value of to know the value of . So I’ll put in equation (3) to get
After simplifying it, I get
Now equation (3) is only true for any value of greater than . So I’ll put in . Therefore I can say that
And this is because
Next, I’ll do the integration. So I can say that
where is the integration constant.
Thus my next step will be to get the values of and respectively.
So I’ll put in equation (6) to get the value of as
Now I’ll simplify it to get
Next, I’ll put in equation (5) to get the value of as
Then I’ll simplify it and that gives
Now I’ll put in equation (4) to get the value of as
Next, I’ll simplify it.
So I can say that
Hence I can conclude that this is the solution to the given example.
Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!