Hi I have problems integrating two questions, how do I integrate sqrt(x^2-4) / x? And (4lnx) / (x(1+(lnx)^2)? Thank you very much!!

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- Dec 6th 2012, 12:15 AMTutuintegrate!
Hi I have problems integrating two questions, how do I integrate sqrt(x^2-4) / x? And (4lnx) / (x(1+(lnx)^2)? Thank you very much!!

- Dec 6th 2012, 04:33 AMcoolgeRe: integrate!
For the second problem, first use t= ln(x), then use z = 1+ t^2. Hope this helps!

- Dec 6th 2012, 08:36 AMSorobanRe: integrate!
Hello, Tutu!

Quote:

$\displaystyle \int\frac{\sqrt{x^2-4}}{x}\,dx$

It should be obvious that a Trig Substitution is called for.

Let $\displaystyle x \,=\,2\sec\theta \quad\Rightarrow\quad dx \,=\,2\sec\theta\tan\theta\,d\theta \quad\Rightarrow\quad \sqrt{x^2-4} \,=\,2\tan\theta$

Substitute: .$\displaystyle \int \frac{2\tan\theta}{2\sec\theta}(2\sec\theta\tan \theta\,d\theta) \;=\;2\int\tan^2\theta\,d\theta $

. . . . . . . . $\displaystyle =\;2\int(\sec^2-1)\,d\theta \;=\;2(\tan\theta - \theta) + C$

Now back-substitute.

- Dec 6th 2012, 08:38 AMHallsofIvyRe: integrate!
- Dec 6th 2012, 01:23 PMtopsquarkRe: integrate!