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

• Sparky_
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.
Sparky_
Greetings,

I know $$\int \frac {1} {{\sqrt{1-x^2}} dx$$ is arcsine(x)

My question is can $$\int \frac {1} {\sqrt{1-x^2}} dx$$
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_

Last edited:
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.

Antiderivatives are unique up to a constant. Ie, all antiderivatives of that will be arcsine(x)+C.

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.

One could certainly express arcsin in terms of logs using Euler's formula...

One could certainly express arcsin in terms of logs using Euler's formula...

...or in terms of a complex logarithm

So yes I think the OP is happy,

possibly other functions (ln for example)[\QUOTE]

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_

If trig substitution, which just about everyone mentioned, is not a "clever integration", I don't know what is!

I agree trig substitution is clever, I was hoping for a solution that doesn't include trig functions.

-Sparky_

I remember what sparky wanted from his previous thread that i tried to help him in. He had $$\arcsin x = -i \ln ( ix \pm \sqrt{1-x^2})$$ as he should, but could not prove why the expressions were equal.

Last edited:
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_

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

$$\arcsin x = \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}$$

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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Integration technique is important because it allows us to solve a wide variety of mathematical problems, including calculating areas, volumes, and finding the average value of a function. It is also a fundamental tool in physics, engineering, and other scientific fields.

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There are several types of integration techniques, including substitution, integration by parts, partial fractions, trigonometric substitution, and numerical integration. Each technique is useful for different types of functions and can be applied in various situations.

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The integration technique to use depends on the type of function and the problem at hand. Some techniques work better for certain types of functions, while others may require multiple techniques to solve. It is important to understand the properties and rules of each technique in order to determine the best approach.

## 5. Can integration technique be used in real-world applications?

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