1. ## recursive integration

hey there,
this is my first post and I am a little concerned regarding notation, but I'll do my best to be as clear as possible in asking my question.

I am trying to find the recursive formula for evaluating the integral of In = [tangent x]^n

The book I am studying with gives as solution this:[1/(n -1 )]*[tangent x]^(n - 1) - I(n-2).

What I am getting, however, is something different. The way I am working is this:

I write [tangent x]^n as [tangent x]*[tangent x]^(n - 1). I proceed finding the derivative of the second multiplier and the integral of the first and proceed with the common steps of integration in parts method.

Now, since I get something different from what the book gives, I suspect two things. I either am rewriting the given function wrongly, and therefore I am getting a different answer, OR, the solution in the book is wrong.

I am not looking for someone to solve this problem for me, but I would appreciate it if someone could help me clarify if I am parting the function wrongly, or perhaps the solution given in the book is wrong?

thank you!

2. ## Re: recursive integration

Originally Posted by bibiki
I am trying to find the recursive formula for evaluating the integral of In = [tangent x]^n

The book I am studying with gives as solution this:[1/(n -1 )]*[tangent x]^(n - 1) - I(n-2).
Hint: write $(\tan x)^n = (\sec^2x-1)(\tan x)^{n-2}.$ No need for integration by parts.

3. ## Re: recursive integration

Thank you Opalg,
I retried the assignment with your hint and it worked. Thanks again.

4. ## Re: recursive integration

Originally Posted by bibiki
hey there,
this is my first post and I am a little concerned regarding notation, but I'll do my best to be as clear as possible in asking my question.

I am trying to find the recursive formula for evaluating the integral of In = [tangent x]^n

The book I am studying with gives as solution this:[1/(n -1 )]*[tangent x]^(n - 1) - I(n-2).

What I am getting, however, is something different. The way I am working is this:

I write [tangent x]^n as [tangent x]*[tangent x]^(n - 1). I proceed finding the derivative of the second multiplier and the integral of the first and proceed with the common steps of integration in parts method.

Now, since I get something different from what the book gives, I suspect two things. I either am rewriting the given function wrongly, and therefore I am getting a different answer, OR, the solution in the book is wrong.

I am not looking for someone to solve this problem for me, but I would appreciate it if someone could help me clarify if I am parting the function wrongly, or perhaps the solution given in the book is wrong?

thank you!
Hello and welcome!

First of all, I think you should separate for cases: i) n is even. ii) n is odd.

Second, in each case use the identity $\tan^2(x)=\sec^2(x)-1$.

EDIT:

Seems bit strange, but I didn't saw your post Opalg, Sorry.

5. ## Re: recursive integration

Zarathustra,
thank you for your reply. I already solved the assignment. however, I don't see why would you want to separate cases for n even and odd. I know you think of positive/negative values of the function, but I don't think that matters in this case?!?!

6. ## Re: recursive integration

Originally Posted by bibiki
Zarathustra,
thank you for your reply. I already solved the assignment. however, I don't see why would you want to separate cases for n even and odd. I know you think of positive/negative values of the function, but I don't think that matters in this case?!?!

If n is even the the 'last' integral that you left to evaluate is $\int (\sec^2(x)-1)dx$ and if n is odd the the 'last' integral that you left to evaluate is $\int \tan(x)(\sec^2(x)-1)dx$