# Homework Help: Integration using an Abel transform

1. May 10, 2013

### |mathematix|

1. The problem statement, all variables and given/known data

Find the following integral:

2. Relevant equations

$$\int \frac{e^{x}}{\sqrt{(1+e^{2x})(1-e^{4x})}}dx$$

3. The attempt at a solution

I changed the integral to: $$\int \frac{e^{x}}{(1+e^{2x})\sqrt{(1-e^{2x})}}dx$$
The let u=e^x
The integral becomes: $$\int \frac{du}{(1+u^{2})\sqrt{(1-u^{2})}}$$
I can do this the long way, such as on wolfram alpha but I want to use an Abel transform so let $$u=\sqrt{1-u^{2}}'$$

$$\sqrt{1-u^{2}}'=-\frac{u}{\sqrt{1-u^2}} \therefore v^{2}=\frac{u^{2}}{1-u^{2}}$$

$$du=\frac{dv}{\sqrt{1-u^{2}}}$$

The integral becomes: $$\int \frac{dv}{1-u^{4}}$$

I need to somehow get rid off the u and get the integral in terms of v so how can I do that?

2. May 10, 2013

### haruspex

u2 = 1 - v2, no?

3. May 10, 2013

### |mathematix|

How do you get that?

4. May 10, 2013

### haruspex

Maybe I misunderstood your substitutions. This doesn't seem to be consistent:
Did you mean $$v=\sqrt{1-u^{2}}'$$? If so, u2 = v2/(1+v2)

5. May 10, 2013

Thanks :)