Integrate Tan^n (x). Hello friends, today I’ll talk about how to integrate Tan^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
- Integrate Cot^(n) x dx | Reduction formula for Cot^(n) x
- Reduction formula for Sec^(n) x | Integrate Sec^(n) x dx
Reduction formulas for logarithmic functions
Reduction formulas for hyperbolic functions
Integrate Tan^n (x)
Solved example of how to integrate Tan^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)*, “By writing as , obtain a reduction formula for
Hence show that
Now here I have to get a reduction formula for . Also I have to prove that
Therefore I’ll start with the integration of .
As I can see from the example itself, I’ll write as
Also, from the standard formulas in trigonometry, I know that
Therefore I can say that
So the reduction formula for will be
Then I’ll simplify it as
And that gives
Now I’ll integrate . Thus it will be
And that means
where is the integration constant. So this gives
Therefore will be
If I say, , then it will be
Also, if , then
And this is the answer to the first part of this example.
Next, I’ll show that .
Thus I’ll substitute the limits in equation (1). So it will be
Also, I already know that
So will be
Then I’ll simplify it to get
Therefore I have also proved the relation in the second part as well.
Hence I can conclude that these are the solutions 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!!