(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is the problem I was given:

[tex]I_{n} = \int^{0}_{\infty}(1 + x^{2})^{-n} dx[/tex]

I was told to "deduce that"

[tex]I_{n} = 2n(I_{n} - I_{n + 1})[/tex]

so I can "Hence or otherwise show that"

[tex]\int^{0}_{\infty}(1 + x^{2})^{-4} dx = \frac{5\pi}{32}[/tex]

2. Relevant equations

I don't even know what I am being asked to do. I have relegated myself to failing this problem. Originally, I figured I would just try to find the original function of the integrand and see if that left me somewhere that made more sense, but I can't find anything in my notes that explains how to solve this with two variables. I am not looking for an answer, but rather maybe a hint as to where I should be looking for an integration technique.

3. The attempt at a solution

The first thing that I gather is that I have to separate the parts, so I could do something like: if

[tex] z = 1 + x^{2} [/tex]

then,

[tex] dz = 2x dx [/tex]

and,

[tex]I_{n} = \int^{0}_{\infty}z(x)^{-n} dz [/tex]

That obviously looks like something that came out of the chain rule, so I first go backwards on the power using [tex] \int a^{x} = \frac{a^{x+1}}{x+1} [/tex]

which combined with the chain rule fives me,

[tex]\int z(x)^{-n} dz = \frac{z(x)^{1-n}}{1-n}[/tex]

So far, so good. I know the integral of [tex](1 + x^{2})[/tex] is [tex](x + \frac{x^{3}}{3})[/tex]

and here is where it all falls apart. I have no idea how to put those two parts together, and I don't know what to review / re-read to figure it out. Can anyone just at least tell me what kind of problem this is so I know what I am supposed to be searching for? As an econ student my calc background is very basic, we never had to deal with this sort of thing.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integrating a two variable equation

**Physics Forums | Science Articles, Homework Help, Discussion**