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Mathematics 88 Online
OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (mathmagician):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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.

OpenStudy (anonymous):

i did u= x+1

OpenStudy (anonymous):

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

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