Here is one I found implication but not by itself
We first do this by seeing that
so
This brings a much simpler calculations of
Because
So we have
I am actually going to use induction on this one, so if I mess it up please forgive me.
Base Case
Let us start with the base case
So we have proved the base case.
Inductive Hypothesis
Now lets assume that
This being our inductive hypothesis (IH)
Inductive step
Now lets look at our inductive step
Which by Integration by Parts
Thus our inductive step is proven.
We can make a stronger case of this
This is gotten by simple aglaebraic manipulation
Now on the left hand integral if we let
We get that
and as \
And as
So we get
Mathstud28, it a really good work! I find it fantastic how you evaluate integrals
(just a small note: the inverse functions of the hyperbolic functions are the area-functions, not arcus functions (as you know of course), which the abbr. name is arsinh, arcosh, artanh, arcoth, arsech, arcsch of - all I mean is, it's not arc... but ar... )
Inverse hyperbolic function - Wikipedia, the free encyclopedia
I pretty much like the notaions arc and ar instead of that one with ^-1 cuz the 2nd one is ambiguous - you can think of it as 1/function
best regards, marine
This one is the one that has caused me the most mental anguish. I had nowhere to start and I just layed in bed and thought about it for twenty minutes and then went back to the drawing board. ad infinitum. It was easy once I realized something.
If we make a substitution
Let
Then as
and as
So we get
So now using this we see the following
Now I wanted so I did the following
Now for the first integral, if you let
you get
Now once again for this first integral let
And as
And as
Giving us
Now seeing that since
Which means that
So
So FINALLY we can conclude that
That one was a toughie.
We know that
and
We see that
Now let and
Then we have
Now when we evaluate this by parts I am sure since I have shown it many times that we get
Now we want to back sub to get
Now let us consider the two cases if
Then we have that
Now if
This implies that
So by the oddness of arctangent
Now if
Then we go back to the beginning and rewite it as this
Which we proved in another thread is equal to
Giving us
Putting htis all together we get