Calculus II - Lengths of Curves - Hard

[GreenPrint]

Homework Statement

Find the arc length of the following curves on the given interval by integrating with respect to x

y=x^4/4+ 1/(8x^2); [1,2]

Homework Equations

Let f have a continuous first derivative on the interval [a,b]. The length of the curve from (a,f(a)) to (b,f(b)) is

L = integral[a,b] sqrt(1+f'(x)^2) dx

The Attempt at a Solution

y=x^4/4+ 1/(8x^2); [1,2]
I took the derivative and got

dy/dx = x^3 - 1/(4x^4)

I than square this and got

x^6 - 1/2 + 1/(16*x^6)

plunging into the formula I get

integral[1,2] sqrt(1/2 + x^6 + 1/(16*x^6)) dx

I have no idea were to go from here. I tried making several substitutions and than realized that there was nothing in my techniques of integration to evaluate this in any way or form. I plugged the indefinite integral into wolfram alpha to see what it got for the antiderivative and apparently it doesn't know how to integrate this neither.
http://www.wolframalpha.com/input/?i=integral+sqrt%281%2F2%2Bx^6%2B1%2F%2816x^6%29%29
well apparently it came up with a antiderivative it just has no steps to prove it to be correct when you click on show steps "(step-by-step results unavailable)". So I conclude that there is something really simple that I'm not seeing or something really complicated. This came from my textbook in -Applications of Integration-Lengths of Curves

Thanks for any help!

 

[Bohrok]

Try factoring 1/(16x⁶) out of x⁶ - 1/2 + 1/(16*x⁶) first.

 

[GreenPrint]

x^6 - 1/2 + 1/(16*x6) = 1/(16*x^6)[16x^12-8x^6+1]
plugging into L = integral[a,b] sqrt(1+f'(x)^2) dx
yields
integral[1,2] sqrt(1+1/(16*x^6)[16x^12-8x^6+1])dx

wasn't sure what to do at this point so I wrote it all as the same fraction

integral[1,2] sqrt((16x^6+16x^12-8x^6+1)/16x^6)dx

simplified -> 1/sqrt(16x^6) = 1/(4x^3)

1/4 integral[1,2] sqrt(16x^6+16x^12-8x^6+1)/x^3 dx

I'm not sure how to proceed from here

 

[Bohrok]

I copied down the wrong part I meant to say factor 1/(16x⁶) from x⁶ - 1/2 + 1/(16*x⁶)
You basically already did it though, so you would get sqrt(1/(16*x⁶)[16x¹²+8x⁶+1])

What can you do with 16x¹²+8x⁶+1?

 

[GreenPrint]

thanks i got it factor it and then it's easy

 

[Ray Vickson]

If your y is *exactly* as written [y = x^4/4 + 1/8/x^2 --- note that a/b/c means a/(b*c) ], then your derivative is wrong, so you have the
wrong integral. The last term in dy/dx is -1/4/x^3, not your -1/4/x^4.

RGV