Integral: sqrt(1+((x^4-1)/(2x^2))^2)

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In summary, the integral of sqrt(1+((x^4-1)/(2x^2))^2) can be simplified using the substitution method. By letting u=(x^4-1)/(2x^2), the integral can be rewritten as sqrt(1+u^2) du, which can then be solved by using trigonometric substitution. This method allows for the integration of complex expressions involving square roots and polynomials.
  • #1
Alexx1
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http://integrals.wolfram.com/index.jsp?expr=sqrt(1%2B((x^4-1)%2F(2x^2))^2)&random=false

Can someone explain me how to solve an integral like this?
I tried it, but I have absolutely no idea how to solve it..
 
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  • #2
Welcome to PF!

Hi Alexx1! Welcome to PF! :smile:
sqrt(1+((x^4-1)/(2x^2))^2)
Alexx1 said:
http://integrals.wolfram.com/index.jsp?expr=sqrt(1%2B((x^4-1)%2F(2x^2))^2)&random=false

Can someone explain me how to solve an integral like this?
I tried it, but I have absolutely no idea how to solve it..

Do you mean [tex]\sqrt(1+(\frac{x^4-1}{2x^2})^2)[/tex] ? :confused:
 
  • #3


tiny-tim said:
Hi Alexx1! Welcome to PF! :smile:



Do you mean [tex]\sqrt(1+(\frac{x^4-1}{2x^2})^2)[/tex] ? :confused:

Yes, it's that integral ;)
 
  • #4
Alexx1 said:
Yes, it's that integral ;)

ok, then (without integrating) expand everything inside the √, and then simplify …

what do you get? :smile:
 
  • #5
tiny-tim said:
ok, then (without integrating) expand everything inside the √, and then simplify …

what do you get? :smile:

(8x^8+2x^4+x)/(4x^4)
 
  • #6
(try using the X2 tag just above the Reply box :wink:)
Alexx1 said:
(8x^8+2x^4+x)/(4x^4)

uhhh? :confused: not even close :redface:
 
  • #7
tiny-tim said:
(try using the X2 tag just above the Reply box :wink:)


uhhh? :confused: not even close :redface:

((x^4-1)/2x^2)^2) = ((x^4-1)^2)/((2x^2)^2)

= (x^8+1-2x^4)/(4x^4)

==> 1 + (x^8+1-2x^4)/(4x^4) = (4x^4+x^8+1-2x^4)/(4x^4)

= (x^8+2x^4+1)/(4x^4)

This is the correct answer?
I made a mistake it had to be: x^8 (instead of 8x^8) and 1 (instead of x)
 
  • #8
(please use the X2 tag just above the Reply box … it makes it much easier to read)
Alexx1 said:
… (x^8+2x^4+1)/(4x^4)

ok! :smile:

and the square-root of that is … ?​
 
  • #9
What should've made factorising easy to spot is that you first expanded
[tex](x^4-1)^2=x^8-2x^4+1[/tex]

and after adding [itex]4x^4[/itex] you now have
[tex]x^8+2x^4+1[/tex]

Isn't it clear how this can be factored?
 
  • #10
The crucial point is that [itex]4a+ (x- a)^2= 4a+ x^2- 2ax+ a^2[/itex][itex]= x^2+ 2ax+ a^2= (x+a)^2[/itex].

That's used a lot to produce easy "arc length" problems!
 
  • #11
tiny-tim said:
(please use the X2 tag just above the Reply box … it makes it much easier to read)ok! :smile:

and the square-root of that is … ?​

= √x8+2x4+1/√4x4

= √(x4+1)2/2x2

= x4+1/2x2
 
  • #12
Alexx1 said:
But now I don't know what to do with : √x8+2x4+1

come on, think! :smile:

(or put x4 = y :wink:)
 
  • #13
tiny-tim said:
come on, think! :smile:

(or put x4 = y :wink:)

Ok, I found it!

= √x8+2x4+1/√4x4

= √(x4+1)2/2x2

= x4+1/2x2

==> Integral x4+1/2x2

= (1/2) Integral (x4/x2) + (1/2) Integral (1/x2)

And so on..
 
  • #14
Thanks everyone for helping me out!
 

Related to Integral: sqrt(1+((x^4-1)/(2x^2))^2)

1. What is the integral of sqrt(1+((x^4-1)/(2x^2))^2)?

The integral of sqrt(1+((x^4-1)/(2x^2))^2) is a complex integral that cannot be expressed in terms of elementary functions. It can be approximated using numerical methods or solved using specialized techniques such as elliptic integrals.

2. How do you solve the integral sqrt(1+((x^4-1)/(2x^2))^2)?

The integral sqrt(1+((x^4-1)/(2x^2))^2) can be solved using specialized techniques such as elliptic integrals or by approximating it using numerical methods. It cannot be solved using traditional methods of integration.

3. What is the purpose of the sqrt(1+((x^4-1)/(2x^2))^2) integral?

The integral sqrt(1+((x^4-1)/(2x^2))^2) is often used in physics and engineering to calculate the arc length of a curve. It can also be used in other applications such as calculating the surface area of certain shapes.

4. Can the integral sqrt(1+((x^4-1)/(2x^2))^2) be simplified?

No, the integral sqrt(1+((x^4-1)/(2x^2))^2) cannot be simplified using traditional methods of integration. It can only be approximated using numerical methods or solved using specialized techniques.

5. Is there a way to express the integral sqrt(1+((x^4-1)/(2x^2))^2) in terms of known functions?

No, the integral sqrt(1+((x^4-1)/(2x^2))^2) cannot be expressed in terms of elementary functions such as polynomials, trigonometric functions, or exponential functions. It can only be approximated using numerical methods or solved using specialized techniques.

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