Ok so:

Its a 3 part question. I've completed parts 1 and 2.

Part 1)

Draw $\displaystyle tanhx $ and $\displaystyle tanh^-1 x $ on the same graph

Part 2)

Find $\displaystyle INT tanh x dx$ between limits k and 0 (where k > 0) which gives you $\displaystyle ln coshk$

Part 3) though...

I need to show $\displaystyle I tanh^-1 x dx$ between tanhk and 0 is $\displaystyle ktanhk - ln coshk$

If you integrate it by parts you form: $\displaystyle [xtanh^-1 x] tanhk -> 0 = ktanhk$ and I t hink $\displaystyle - int x/(1-x^2)$ between the same limits. I can't work out how to link lncoshk and that integral. And with the previous parts in the question.

Thanks in advance!

P.S How do you do an integral sign on this forum :p