Question

integral of 1/ ((x) * sqrt(x+1))) dx

Answer 1

this is a very difficult integral so if someone can help me I would appreciate it

Answer 2

The answer is \[\int\limits_{?}^{}1/(x \sqrt{1+x})dx=-2*arctanh(\sqrt{x+1)}\]

Answer 3

Ya i figure that out, helllla hard to do though!

Answer 4

It's not hard. Substitute \(u=\sqrt{x+1} \implies x=u^2-1 \implies dx=2udu\). That gives:
\[\int\limits_{}^{}{dx \over x \sqrt{x+1}}=\int\limits {2u du \over (u^2-1)(u)}=\int\limits {2 \over u^2-1}=-2\int\limits {1 \over 1-u^2}=-2\tanh^{-1}u.\] Now, substitute back for x. You can also solve from the last step by using partial fractions.

Answer 5

i did u= x+1

Answer 6

That would work, but you will need another substitution, say \(z=\sqrt{u}\).