You are better at this than me. That's for sure. I have to ask. How in the world did you get that ln thing from that integral with arctan?.
I can do each of those integrals you have, but I never saw it in terms of ln.
I started out using the series for arctan, but the x^2+1 in there caused me a fit.
I can see , which
when multiplied by the existing 2k+1 will result in the Catalan, but I am
getting hung up because of the x^2+1.
Then we can write
When we integrate, we have
When multiplied with , we get the Catalan.
But what about the rest?. I know it;s there. I can smell it.
I am making a stupid mistake. I just know it. Every time I tried this thing I ended up in a dead end.
You probably see it right off, though.
Oh, OK. Good egg. I see now what you done. Clever indeed.
I also know that
Which is part of our solution. May have to look into how to transform it. I bet there is a connection.
Also,this integral becomes , but the Pi/4 makes it more difficult than the usual Pi/2 for this integral.