Can the Integral of 1/sqrt(1-x^2) be Solved without Using Trig Functions?

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In summary, Gib_Z explained that arcsin(x) can be solved by using a trig substitution and that the only other way to write the antiderivative is in terms of a power series or a complex logarithm. However, if the problem is in the real world, then these methods won't work. Antiderivatives are unique up to a constant.
  • #1
Sparky_
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Greetings,

I know [tex] \int \frac {1} {{\sqrt{1-x^2}} dx[/tex] is arcsine(x)

My question is can [tex] \int \frac {1} {\sqrt{1-x^2}} dx[/tex]
be solved by some technique (parts, substitution...) and the answer be in terms of x and NOT an expression of arcsine?

Meaning, I would like the solution to the integral in terms of x and possibly other functions (ln for example) but not in terms of trig functions.

If so, can you show me?

Thanks
Sparky_
 
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  • #2
It can be done by an appropriate trig substitution ie. x=sin(y), but the answer will end up identically. I'm fairly certain that arcsin(x) is the only answer you will ever get though.
 
  • #3
Antiderivatives are unique up to a constant. Ie, all antiderivatives of that will be arcsine(x)+C.
 
  • #4
The only two other (sane) ways I can think of to write that antiderivative is as a power series, or in terms of a complex logarithm. But any of those ways will simply be a different way of writing arcsin x.
 
  • #5
One could certainly express arcsin in terms of logs using Euler's formula...
 
  • #6
DeadWolfe said:
One could certainly express arcsin in terms of logs using Euler's formula...

Yea, Hurkyl already mentioned that.

...or in terms of a complex logarithm

So yes I think the OP is happy,

possibly other functions (ln for example)[\QUOTE]
 
  • #7
What I am wanting is an expression for arcsine that uses other functions (hopefully LN and powers of x) but I am hoping to work in the real domain.

Gib Z help me quite a bit with a derivation regarding arcsine.

I was hoping some clever integration could come up with another expression equal to arcsin by integrating the derivative of arcsine.

Any thoughts?

Thanks
Sparky_
 
  • #8
If trig substitution, which just about everyone mentioned, is not a "clever integration", I don't know what is!
 
  • #9
I agree trig substitution is clever, I was hoping for a solution that doesn't include trig functions.

-Sparky_
 
  • #10
I remember what sparky wanted from his previous thread that i tried to help him in. He had [tex]\arcsin x = -i \ln ( ix \pm \sqrt{1-x^2})[/tex] as he should, but could not prove why the expressions were equal.
 
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  • #11
Correct Gib_Z and you did help me prove it - you helped me get the true solution and in that I ended up with some other solutions that I assumed involved the other quadrants.

Using that expression (arcsin = -iln(...) has become very involved due to working in the complex domain.

I'm now searching for a similar type of solution in the real domain only. (I suppose this rules out Euler's)

-Sparky_
 
  • #12
Well then the best answer I can give you is the Taylor Series for Arcsin x, which If i remember correctly is this:

[tex]\arcsin x = \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}[/tex]
 
  • #13
Thank you all for your help.

Sparky_
 

1. What is integration technique?

Integration technique is a method used in mathematics to find the integral of a function. It involves finding the area under the curve of a function by breaking it down into smaller, simpler parts and then adding them together.

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