A Few Integrals

**GreySweatshirt306** · February 12th 2009, 09:49 PM

I'm stumped on these:

The integral of x^3 (1+x^2)^3/2

The integral of the square root of (x^2 - 1)dx/x^3

The integral of x^3 dx/(4-x^2)^2

**Jhevon** · February 12th 2009, 09:53 PM

Originally Posted by GreySweatshirt306

I'm stumped on these:

The integral of x^3 (1+x^2)^3/2

do a substitution, $u = 1 + x^2$

The integral of the square root of (x^2 - 1)dx/x^3

note that $\frac {x^2 - 1}{x^3} = \frac {x^2}{x^3} - \frac 1{x^3} = \frac 1x - x^{-3}$

The integral of x^3 dx/(4-x^2)^2

another substitution problem, let $u = 4 - x^2$

**GreySweatshirt306** · February 12th 2009, 10:04 PM

Thanks for the help--but I don't see how a substitution works for either of those two problems.

**Jhevon** · February 12th 2009, 10:39 PM

Originally Posted by GreySweatshirt306

Thanks for the help--but I don't see how a substitution works for either of those two problems.

did you try it?

$\int {x^3}{(1 + x^2)^{3/2}}~dx$

Let $u = 1 + x^2 \implies \boxed{x^2 = u - 1}$ ....note that $x^3 = x \cdot x^2$

$\Rightarrow du = 2x~dx$

$\Rightarrow \frac 12 ~du = x~dx$

Thus our integral becomes

$\frac 12 \int \frac {u - 1}{u^{3/2}}~du$

i suppose you can take it from here. the other integral is similar

**GreySweatshirt306** · February 17th 2009, 06:45 AM

Is there anyway to do these using a trig sub?

**Krizalid** · February 17th 2009, 07:15 AM

Utterly unnecessary.

1) $x=\tan\varphi.$

2) $x=\sec\varphi.$

3) $x=2\sin\varphi.$

**GreySweatshirt306** · February 17th 2009, 09:47 AM

Thanks for the help. I'm not very good at these though, could you flesh them out a bit? I really appreciate anything you can do

**Jhevon** · February 17th 2009, 04:45 PM

Originally Posted by GreySweatshirt306

Thanks for the help. I'm not very good at these though, could you flesh them out a bit? I really appreciate anything you can do

what's wrong with the way they were done before? are you required to use trig subs?

**GreySweatshirt306** · February 17th 2009, 05:35 PM

Trig Subs are required

**Krizalid** · February 17th 2009, 05:37 PM

You didn't specify that from the beginning, and that's bad, pretty bad since you wasted Jhevon's time.

If you want to be helped, show at least what you've tried.

**GreySweatshirt306** · February 17th 2009, 05:53 PM

Well let me edit that--
Trig subs are NOT required, what I meant to say is that I came to my math teacher with some of the work that Jhevon got me started on (thanks again), and he instructed me to also try to work them out using a trig substitution, so i'm back here. Sorry for the confusion

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