How to Solve This Complex Trigonometric Integral?

[TheFerruccio]

Homework Statement

Evaluate the integral.

Homework Equations

$$\int{\arccos{\frac{a}{a+\alpha}}\sqrt{\frac{(a+\alpha)^2}{(a+\alpha)^2-x^2}}d(a+\alpha)}$$

For reference, this is the solution, but I do not know how to get here:

$$\frac{a}{2}\ln{\frac{\xi+1}{\xi-1}} -\frac{x}{2}\ln{\frac{\xi+\frac{x}{a}}{\xi-\frac{x}{a}}}$$
where
$$\xi^2=\frac{(a+\alpha)^2-x^2}{(a+\alpha)^2-a^2}$$

The Attempt at a Solution

First step for me was to integrate by parts. I set $$c=a+\alpha$$ and integrated using c as my working variable.

After integrating by parts once, I end up with:

$$\left[\arccos{\frac{a}{c}}\sqrt{c^2-x^2}\right]_{boundary}-\int{\frac{a}{c}\sqrt{\frac{c^2-x^2}{c^2-a^2}}dc}$$

I am not sure what to do here. I was thinking of trying some sort of u substitution, maybe having $$u=\sqrt{\frac{c^2-a^2}{c^2-x^2}}$$.

There are additional restrictions on how these all relate to each other. For instance:

c > 0
a > 0
α > 0
c=a+α

Perhaps they can assist me with further limiting the scope of this integral and making it evaluate. Right now, if I try to put this integral into Mathematica, I end up with a statement which includes the Appell Hypergeometric function.

It should be noted that I also do not know what the limits of integration would be. Perhaps having the solution would give some insight into what the limits of integration are, but I do not see it. I have generated a stack of paper over the course of a week trying to figure out this integral, and I am getting absolutely nowhere. Some further assistance would be fantastic.

As a side note: Please tell me if my questions are being somehow vague on these forums. I would greatly like to improve the clarity of my questions for others. Given that every thread I have created in the past several months on this forum has been completely devoid of replies, I have to wonder whether I am fundamentally missing some key piece of information in my descriptions that results in scaring at the potential help away.

 

[jedishrfu]

Have you tried working backward from the answer to see if you can see how to proceed?

 

[Mark44]

For this integral -- $\int \frac{a}{c}\sqrt{\frac{c^2-x^2}{c^2-a^2}}dc$ -- I'd be more inclined to try a trig substitution.

 

[TheFerruccio]

Both of these are very good ideas. I will see if I can do these. I went to the professor and he said that it's probably in a table somewhere, so I do not think he did the algebra either. After seeing a square root of squares, I did default to thinking it must be some kind of triangle equality I could set up.

 

[Mark44]

Any time you have a sum or difference of squares, or the square root of a sum or difference of squares, trig substitution is a good strategy. Keep in mind here that x is kind of a red herring - the variable of integration is c.