Integral with a radical on top

[airforceone]

Hi,

I've been trying to figure out how to integrate this integral:

I used trig substitution, and this is what I have so far:

I know that the integral for tangent-squared is in the back of the book, but how can I solve this using just the basic integrals/derivatives that we are to memorize? I tried integration by parts but the resulting integral would be complicated (there would be a secant-squared).

Thanks and I'm glad this forum is still around after all these years!

 

[soroban]

Hello, airforceone!

\(\displaystyle \int^2_1\frac{\sqrt{x^2-1}}{x}\,dx\)

I used trig substitution, and this is what I have so far:

. . . \(\displaystyle \int\tan^2\tgheta\,d\theta\) . . Right!

\(\displaystyle \text{Use }\:\tan^2\theta \:=\:\sec^2\theta-1\)

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

Got it?

 

[airforceone]

Thanks! I re-looked at the table of integrals that we should memorize and secant-squared actually was on it.