Integrating the sqrt of a function

[ck99]

Homework Statement

Apologies for the vague title, I'm not reall sure what I'm look at here! I am doing some revision on solving the Friedmann equations, and in a lot of cases I end up having to integrate a function that looks like

(xaⁿ - 1)^-1/2 da = dt

where a is a function of t, x is a constant, and I am trying to find an equation that describes a as a function of t. The first examples I have found have n = -1, n = -2

Homework Equations

The Attempt at a Solution

I do have a solution for one of these in my lecture notes. In the case where

dt = (xa^-1-1)^-1/2

the solution is given as

t = x arctan ((xa^-1-1)^-1/2) - a^1/2 (x - a)^1/2

I would normally look on wolfram and work through their step-by-step solution to teach myself the method, but even wolfram can't answer this one! Any help would be much appreciated.

 

[epenguin]

Homework Statement

Apologies for the vague title, I'm not reall sure what I'm look at here! I am doing some revision on solving the Friedmann equations, and in a lot of cases I end up having to integrate a function that looks like

(xaⁿ - 1)^-1/2 da = dt

where a is a function of t, x is a constant, and I am trying to find an equation that describes a as a function of t. The first examples I have found have n = -1, n = -2

Homework Equations

The Attempt at a Solution

I do have a solution for one of these in my lecture notes. In the case where

dt = (xa^-1-1)^-1/2

the solution is given as

t = x arctan ((xa^-1-1)^-1/2) - a^1/2 (x - a)^1/2

I would normally look on wolfram and work through their step-by-step solution to teach myself the method, but even wolfram can't answer this one! Any help would be much appreciated.

You can check the stated result by differentiating which is often all the insight anyone needs.

 

[ck99]

Thanks for the tip. I have tried that approach, and double-checked my differentiation with wolfram, but I can't get from the given solution to the starting point in that direction either!

I started with the first term in the solution

d/da x arctan ((xa^-1-1)^-1/2)

. . . lots of substitutions . . .

= x²(1+(xa^-1-1)^-1/2)^-1(2a²(xa^-1-1)^3/2)^-1

Phew! And for the second term in the solution, I got

d/da a^1/2 (x - a)^1/2 = (x-2a)(2(xa-a²)^1/2)^-1

As I say, I have checked these differentiations with Wolfram so I am sure they are correct (although there is always the possibility of transcription errors when typing such complex formulae) but I can't see how they add up to give the original expression I was trying to integrate. What could I try next?

 

[epenguin]

Thanks for the tip. I have tried that approach, and double-checked my differentiation with wolfram, but I can't get from the given solution to the starting point in that direction either!

I started with the first term in the solution

d/da x arctan ((xa^-1-1)^-1/2)

. . . lots of substitutions . . .

= x²(1+(xa^-1-1)^-1/2)^-1(2a²(xa^-1-1)^3/2)^-1

Phew! And for the second term in the solution, I got

d/da a^1/2 (x - a)^1/2 = (x-2a)(2(xa-a²)^1/2)^-1

As I say, I have checked these differentiations with Wolfram so I am sure they are correct (although there is always the possibility of transcription errors when typing such complex formulae) but I can't see how they add up to give the original expression I was trying to integrate. What could I try next?

I don't have time today to go through that - perhaps another site member could. Showing more of your working wouldn't harm.

Excercise for students!

Are you sure you quoted result right? Some other bracket? For the second part I got

- x a^-1/2 (x - a)^-1/2 .