Calculating the Primitive of sqrt(x^2-4)/x^4 using Substitution Method

[joao_pimentel]

Hi there people

Can anyone help me with this issue?

I'm trying to calculate this primitive

P sqrt(x^2-4)/x^4

I tried the substitution x=2*sec(t) and it seems to work but at the end I get something like:

1/6*(sin(arccos(2/x)))^3

and this is quite different from what we can observe at Wolfram which is:

((x^2-4)^(3/2)) / (12x^3)


Can anyone give me some suggestion?

Thanks in advance!

João

http://MatemáticaViva.pt

 

[arildno]

Well, try to rewrite sin (arccos(y)), using the relation:
$$sin^{2}x+\cos^{2}x=1, x=arccos(y)$$

 

[joao_pimentel]

Thank you very much... it seems to work, but there is any other way to integrate sqrt(x^2-4)/x^4 without trigonometric functions, making it directly without the transformation x=a*sec(t) ?

Thank you

João

http://MatemáticaViva.pt

 

[arildno]

Well, you might use the hyperbolic substitution, x=2cosh(y)

 

[joao_pimentel]

Ok... I will try, but I was trying to figure out how to reach the result ((x^2-4)^(3/2)) / (12x^3) given by Wolfram, which I suppose is correct, without trigonometric functions, since the result does not involve any trigonometry...

 

[joao_pimentel]

Thank you very much

But kindly look at this:
http://www.wolframalpha.com/input/?i=integrate+sqrt(x^2-4)/x^4

There's any way of calculating this integral without using trigonometry?

Thank you in advance

João

 

[arildno]

Not that I know of.

 

[joao_pimentel]

Hi

Thank you very much for your attention...

I already solved this primitive/integral using the suggestion you have given to me...

I used the substitution x=a.sec(t)

You may see the complete results and solution here
http://www.matematicaviva.pt/2010/11/como-calcular-primitiva-de-sqrtx2-4x4.html

Thank you very much again

João