How can one evaluate the integral $$\int\frac{\sqrt{x^2-1}}x\mathrm dx$$?
I tried substituting $x = \cosh t$ but got stuck at $$\int\frac{\sinh^2t}{\cosh t}\mathrm dt$$
Any hints?
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3$\begingroup$ Why not substitute $x = \sec u$? $\endgroup$– user71641Jan 14, 2015 at 20:45
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$\begingroup$ Now use $\sinh^{2}t=\cosh^{2}t-1$ $\endgroup$– user84413Jan 14, 2015 at 20:49
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$\begingroup$ @user84413: I tried that, but then didn't know how to integrate $\frac1{\cosh x}$... $\endgroup$– rubikJan 14, 2015 at 20:50
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$\begingroup$ Here is one way to do it: calc101.com/special_10.html $\endgroup$– user84413Jan 14, 2015 at 22:13
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$\begingroup$ @user84413: Thanks for the reference! That's interesting, although I'll never remember that during an exam! $\endgroup$– rubikJan 15, 2015 at 10:41
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4 Answers
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Let $ x = \sec u $. Then, $ \mathrm{d}x = \sec u \tan u \, \mathrm{d}u $. Then, the integral becomes $$ \int \tan^2 u \, \mathrm{d}u = \int \left( \sec^2 u - 1 \right) \, \mathrm{d}u = \tan u - u + \mathcal{C}. $$ Then, you can substitute back and finish.
answered Jan 14, 2015 at 20:46
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$\begingroup$ Interesting! I am a bit inexperienced with trig integrals. I just know the basic ones, and always forget about, for example, $\int\tan^2x\mathrm dx$! $\endgroup$– rubikJan 14, 2015 at 21:15
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$\begingroup$ @rubik No problem; that happens to everybody! $\endgroup$ Jan 14, 2015 at 21:50
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Put $X^2=x^2-1$ then $x^2=X^2+1$.
$2x\mathrm dx=2X\mathrm dX$.
$$\int\frac{\sqrt{x^2-1}}{x}\mathrm dx=\int\frac{X^2}{x^2}\mathrm dX=\int\frac{X^2}{X^2+1}\mathrm dX$$
answered Jan 14, 2015 at 20:48
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$\begingroup$ Very nice. I actually thought about this substitution, but didn't see that the $x$ would become $x^2$. Damn! $\endgroup$– rubikJan 14, 2015 at 21:04
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$\begingroup$ Thanks. I also though it wouldn't work but who knows let's try! $\endgroup$ Jan 14, 2015 at 21:06
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I'll use the hyperbolic substitution you made. (Why not?) Of importance is the hyperbolic dual of the Pythagorean identity, $\cosh^2 x - \sinh^2 x = 1$. Then, one can see that: $$\frac{\sinh^2 t}{\cosh t} = \frac{\cosh^2 t - 1}{\cosh t} $$
This makes your integral: $$\int \cosh t - \operatorname{sech} t\,dt$$
If you know your hyperbolic trig integrals as well as most people know their "normal" trig integrals, you're home free.
Hint: $$\int\operatorname{sech} t\,dt = 2\arctan\left(\tanh\left(\frac{t}{2}\right)\right) + C$$ (According to Wolfram.)
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$$ \begin{aligned}\int \frac{\sqrt{x^{2}-1}}{x} d x =& \int \frac{x^{2}-1}{x \sqrt{x^{2}-1}} d x \\ =& \int \frac{x^{2}-1}{x^{2}} d\left(\sqrt{x^{2}-1}\right) \\ =& \int\left(1-\frac{1}{x^{2}}\right) d\left(\sqrt{x^{2}-1}\right) \\ =& \sqrt{x^{2}-1}-\int \frac{d\left(\sqrt{x^{2}-1}\right)}{\left(\sqrt{x^{2}-1}\right)^{2}+1} \\ =& \sqrt{x^{2}-1}-\tan ^{-1}\left(\sqrt{x^{2}-1}\right)+C \end{aligned} $$
