<s>[INST] Indefinite intergral of $\int { \sqrt{ x^2-a^2} \over x } \mathrm dx $ I need to integrate $$\int { \sqrt{x^2-a^2} \over x } \mathrm dx $$ using substition, and show it equals $$ \sqrt{x^2 - a^2} - a(\operatorname{arcsec} ({x\over a })) +c $$
I've tried $x=a\sin t$ but I couldn't finish it out. 
Thank in advance for any help. 
 [/INST] Use the substitution: 
$$x = a \sec(\theta)\implies dx = a \sec\theta\tan\theta\,d\theta$$
And use the identity
$$\sec^2(\theta)-1 = \tan^2(\theta).$$
$\require{cancel}$
Then your integral becomes: $$\begin{align}\int { \sqrt{x^2-a^2} \over x } dx & = \int \dfrac{\sqrt{a^2\sec^2 \theta - a^2}(\cancel{a \sec\theta}\tan\theta \,d\theta) }{\cancel{a\sec\theta}}\, \\ \\ & = a\int \sqrt{\tan^2\theta}\, \tan\theta\,d\theta \\ \\ &= a\int \tan^2 \theta \,d\theta \\ \\ & = a\int (\sec^2 \theta - 1)\,d\theta \\ \\ & = a \int \sec^2 \theta \,d\theta - a\int d\theta\\ \\ & = a \tan\theta - a\theta + c\end{align}$$
Now, back substitute to get $$\sqrt{x^2 - a^2} - a\left(\operatorname{arcsec} \left({x\over a }\right)\right) +c$$
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