What is the integral of: 1/(x * sqrt(x^2 + 4))

Question

Hi, Im having a lot of trouble with this calculus problem. I used x= 2 tan theta for my substitution, rather than any u substitution because my professor does not do it that way. However after I plugged that back in and simplified I got 1/2 times the integral of sec theta/ tan theta. When I try using the identity of that as csc and plugging my value of theta as arctan (x/2). However, my online math homework keeps marking it as wrong and online integrator calculator gives a completely different answer, so I'm not sure where I'm going wrong. Thank you so much for your help.

Answer 1

I believe you are on the right track.

You should get to (1/2)∫csc θ dθ which integrates as a logarithm...not easy to get, but check a short table of integrals.