integrate [csch^2(2/x)]/(x^2) dx

I used 'u' substitution with x^-2
which gives me -1/2du=dx/x^3
I know the answer is 1/2coth(2/x)+C
so I think I'm pretty close, I just can't find the 'du' in the integral with the substitution I have....where did I go wrong?

Question

integrate [csch^2(2/x)]/(x^2) dx

I used 'u' substitution with x^-2 
which gives me -1/2du=dx/x^3
I know the answer is 1/2coth(2/x)+C
so I think I'm pretty close, I just can't find the 'du' in the integral with the substitution I have....where did I go wrong?

hartnn · Answer

put u= 2/x
du=.... ?

anonymous · Answer

Without writing that out, I don't think I'll end up with a du to sub back in to put in terms of u..

hartnn · Answer

means ? 
if u=2/x 
du = -2/x^2 dx 
so, 
(1/x^2) dx = du/(-2)
substitute this in your original integral.

anonymous · Answer

wouldn't it be (-1/2)du=(1/x^2)dx?

anonymous · Answer

Didn't see your / ... nevermind my stupidity. Thanks

hartnn · Answer

ok..so now can you solve ?

anonymous · Answer

Yeah, got it now...didn't see (2/x) working in my head...I should really learn to write/test things out. Appreciate the help!

hartnn · Answer

welcome ^_^