Integrating a two variable equation

[groditi]

Homework Statement

This is the problem I was given:

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

I was told to "deduce that"

I_{n} = 2n(I_{n} - I_{n + 1})

so I can "Hence or otherwise show that"

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

Homework 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.

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

z = 1 + x^{2}

then,

dz = 2x dx

and,

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


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

which combined with the chain rule fives me,
\int z(x)^{-n} dz = \frac{z(x)^{1-n}}{1-n}

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

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.

 

[Mark44]

Homework Statement

This is the problem I was given:

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

All of your limits of integration are upside-down. The integrals should look like this:
I_{n} = \int_0^{\infty}(1 + x^{2})^{-n} dx

I was told to "deduce that"

I_{n} = 2n(I_{n} - I_{n + 1})

so I can "Hence or otherwise show that"

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

Homework 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.

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

z = 1 + x^{2}

then,

dz = 2x dx

and,

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

This (above) isn't right. It looks like you just replaced dx with dz, but dz = 2x dx. That 2x has to come from somewhere.

Instead of an ordinary substitution like you tried, a trig substitution might be what is called for.

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

which combined with the chain rule fives me,
\int z(x)^{-n} dz = \frac{z(x)^{1-n}}{1-n}

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

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.

 

[rock.freak667]

Try using integrating by parts with u=(1+x²)^-n.


After which, you will need to use something along the lines of x= x+9-9

 

[groditi]

All of your limits of integration are upside-down. The integrals should look like this:
I_{n} = \int_0^{\infty}(1 + x^{2})^{-n} dx

This (above) isn't right. It looks like you just replaced dx with dz, but dz = 2x dx. That 2x has to come from somewhere.

Instead of an ordinary substitution like you tried, a trig substitution might be what is called for.

Yikes. That was a latex typo, my apologies. Thank you for pointing out that dx error, i was under the impression that the 2xwould come from the derivative of x^{2}. I am learning this all from a set of notes and some videos, all by myself (no teacher or classmates) so it feels very overwhelming.

I thought about the trig substitution too. I remember that:
f(x) = tan^{-1}(x)
and
f'(x) = \frac{1}{1+x^{2}}but I was unable tie that part together to ⁿ part. I get that \frac{1}{(1+x^{2})^{n}} could just be expanded to (\frac{1}{(1+x^{2})})_{n} (\frac{1}{(1+x^{2})})_{n-1} \cdots (\frac{1}{(1+x^{2})})_{0}

but that still leaves me pretty confused.

At the core of it, I recognize that my calculus is poor. In fact, I have absolutely no clue what to do with the "n" term, I've never integrated something with two variables, so I don't know the special rules or formulas it takes to combine them. Looking through the notes I have on integrals, there's nothing with more than one variable in it, so I don't know where to look for more information. is that what they call partial differential equations? If I'm beyond help on this one, how about a suggestion for a textbook?

 

[rock.freak667]

Think of 'n' as a constant rather than a variable. If you try integration by parts, it works out a bit easier than the trig. substitution IMO.

 

[vela]

Oftentimes, when you're asked to deduce a recurrence relation involving integrals, integration by parts, as rock.freak suggested, is a good place to start.