# Integral4

• Aug 6th 2008, 05:06 AM
Apprentice123
Integral4
$\displaystyle \int \frac{dx}{x^4\sqrt{1+x^2}}$

$\displaystyle \frac{(2x^2-1)(1+x^2)^\frac{1}{2}}{3x^3}$
• Aug 6th 2008, 06:20 AM
galactus
There are many ways, but I like trig sub for some reason.

Let $\displaystyle x=tan(u), \;\ dx=sec^{2}(u)du$

When we make the subs we get:

$\displaystyle \int csc(u)cot^{3}(u)du$

$\displaystyle =\int csc(u)(csc^{2}(u)-1)cot(u)du$

Now, can you continue?. Another sub should do it, and when you get to the end remember to let $\displaystyle u=tan^{-1}(x)$ to get it back in terms of x.
• Aug 6th 2008, 06:27 AM
kalagota
also, from same substitution,

$\displaystyle \int \frac{\cos^3 u}{\sin^4 u} \, du$

$\displaystyle t= \sin u$
$\displaystyle dt = \cos u \, du$
$\displaystyle \cos^2 u = 1-\sin^2 u = 1-t^2$

$\displaystyle \int \frac{1-t^2}{t^4} \, dt$
• Aug 6th 2008, 11:05 AM
Krizalid
A reciprocal substitution also works.
• Aug 6th 2008, 04:06 PM
Apprentice123
thanks
• Aug 6th 2008, 06:39 PM
mr fantastic
Quote:

Originally Posted by Krizalid
A reciprocal substitution also works.

I'ts very odd .... I'm certain I posted the same suggestion and some general details (it was post #2) but that post seems to have vanished.
• Aug 7th 2008, 11:14 AM
Krizalid
Dunno. I said it 'cause I saw nobody told it before.