How to use rationalizing substitution on a given equation

Question

How do you use rationalizing substitution to put the equation 2∫√(x/(2(Cx+GM))) into the form k∫√(a² - u²)?

Answer 1

I am not quite sure I understand the question correctly.

I am assuming that C, G, M are given constants and you are supposed to find

constants a, k -- derived from C, G, M, so that

2∫√(x/(2(Cx+GM))) dx = k∫√(a² - u²) du

If my guess is right then I do not understand why the given integral is in such a convoluted way.

If my guess is wrong, I hope you can use my solution to find solution to the correct problem.

The solution below applies only if C≠0.

The solution is

a = √(GM/2), k = 2(-C/2)^3/2

As the solution involves square roots, the constants a, k may be complex numbers.

In case complex numbers are not allowed, then the solution does not apply for some values of C, G, M.

First to simplify the the problem, use the shorthand

c = C/2

m = GM/2

Then

2∫√(x/(2(Cx+GM)))dx = ∫√(x/(cx+m)) dx

Let

u = √(cx+m)

Then

x = (u²-m)/c

dx/du = 2u/c

∫√(x/(cx+m)) dx =

∫(√x / √(cx+m)) dx =

∫(√((u²-m)/c) / u) (2u/c) du =

∫√((u²-m)/c) (2/c) du =

(2/c) ∫√((u²-m)/c) du =

(2/c)√(-1/c) ∫√(m-u²) du

So a = √m, k = (2/c)√(-1/c)