Prove that the integral is equal to ##\pi^2/8##

[fresh_42]

@fresh_42 I found a solution to this integral here. Does it convince you?
I can't understand how they derived equation $(3)$ from equation $(2)$.

Looks nice at first sight. They say how to do it. Looks like a bit of work to check it. I'll go through it if I have some time. $\theta'$ has been my problem, too.

 

[Meden Agan]

Looks nice at first sight. They say how to do it. Looks like a bit of work to check it. I'll go through it if I have some time. $\theta'$ has been my problem, too.

Sure. But if you differentiate $\Theta(t) = \arcsin \left(\sqrt{E(t)}\right)$ with respect to $t$, you don't obtain the expression for $\Theta'(t)$ given by $(3)$.

 

[robphy]

@fresh_42 I found a solution to this integral here. Does it convince you?
I can't understand how they derived equation $(3)$ from equation $(2)$.

Sadly, there's no context.
Where did this integral arise?
What motivated the specific change of variables?
What characterizes the class of similar integrals solved by this method?
From what's presented, it seems it's "guess and check".

 

[pines-demon]

Sadly, there's no context.
Where did this integral arise?
What motivated the specific change of variables?
What characterizes the class of similar integrals solved by this method?
From what's presented, it seems it's "guess and check".

See post #26.

 

[fresh_42]

Sure. But if you differentiate $\Theta(t) = \arcsin \left(\sqrt{E(t)}\right)$ with respect to $t$, you don't obtain the expression for $\Theta'(t)$ given by $(3)$.

And if you do what he wrote, you won't obtain it either. $(2)$ results in a monster polynomial in $t,$ $\sin\theta$ and $\sin 2\theta.$ To claim "one arrives at" seems a bit too optimistic to me. Where does the root come from? It looks as if he put all the trouble I had, and that prevented me from solving it, into these three words. Quite as if he calculated from behind and claimed what he needed to arrive at the correct number.

You are right. What would really be the clue we were looking for is still hidden. I have had all those formulas, too, at one or another stage of my attempts and failed between $(2)$ and $(3).$ To summarize it as "one arrives at" is perky.

 

[fresh_42]

I had something similar:
$$
I=2\int_0^a\left(\int_0^{\sqrt{f(\alpha)f(-\alpha)}} \dfrac{1}{\sqrt{1-t^2}}\,dt\right)\,d\alpha\, , \,a=\operatorname{arcsin}\left(1/\sqrt[4]{8}\right)
$$
The problem is the inner integral limit.