{ 4/[x sqrt(x^2 - 4)] dx

The answer is 2 arcsec (x/2) but can anyone show me the steps to in solving it? I tried partial decomposition and some trig substitution but nothing really worked.

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- Feb 4th 2009, 08:11 AMKaitosanTricky integral
{ 4/[x sqrt(x^2 - 4)] dx

The answer is 2 arcsec (x/2) but can anyone show me the steps to in solving it? I tried partial decomposition and some trig substitution but nothing really worked. - Feb 4th 2009, 09:00 AMJester
Let $\displaystyle x = 2 \sec \theta$ so $\displaystyle dx = 2 \sec \theta \tan \theta d \theta$

and the integral becomes

$\displaystyle \int \frac{4 \cdot 2 \sec \theta \tan \theta }{2 \sec \theta \;2 \tan \theta} d \theta = 2 \int d \theta $

integrate and use your substitution to get your answer - Feb 4th 2009, 09:17 AMKaitosan
OOH I see. I was confused because I wasn't used to substituting any "x" outside trigonometric expressions. Thanks. :)