Using integration by parts to prove reduction fomula

[Ianfinity]

Use integration by parts to prove the reduction formula:
int(sec^n)x dx = (tan(x)*sec^(n-2)*x)/(n-1) + [(n-2)/(n-1)]int(sec^(n-2)*x dx

n /= 1 (n does not equal 1)

I used "int" in place of the integral sign.


This was a problem on the corresponding test from the cal A class I am from the past semester. I think there might be something like it on my test tomorrow, but I can't figure it out. Apologies if my formatting of the problem is confusing.

Thanks in advance.

 

[Mark44]

Use integration by parts to prove the reduction formula:
int(sec^n)x dx = (tan(x)*sec^(n-2)*x)/(n-1) + [(n-2)/(n-1)]int(sec^(n-2)*x dx

n /= 1 (n does not equal 1)

I used "int" in place of the integral sign.


This was a problem on the corresponding test from the cal A class I am from the past semester. I think there might be something like it on my test tomorrow, but I can't figure it out. Apologies if my formatting of the problem is confusing.

Thanks in advance.

What's your question?

Since you're doing integration by parts, it would help if you show us what you used for u and dv.

 

[Ianfinity]

What's your question?

Since you're doing integration by parts, it would help if you show us what you used for u and dv.

The question is how do I prove the reduction formula here.

I used u=(sec x)^n and dv=dx, so du=sec(x)tan(x)dx and v=x

Where I'm getting most confused at is the part where it says to add (n-2)/(n-1)int(sec^(n-2))xdx

I don't see why (n-2)/(n-1) is multiplied by the integral there... I know I'm missing something, but what? Do I need to use substitution rule in here at some point?

Thanks for your timely response! I didn't expect to see anything for a few hours.

 

[jimbobian]

Hi Ianfinity,
In your response, you have said that u=(secx)^n , but having made that choice your expression for du/dx is incorrect. I don't think your choice of u is going to yield any useful results, you need to manipulate it a bit before you can apply parts.

As a hint: When you use parts to find a reduction formula, you will usually be aiming to get the new integral to be your original integral to a different power. You can see from the problem statement that this is included in the final answer (sec(x)^(n-2))
You need to think about how the original integral (sec(x)^n) has been split, to allow this term to be formed.