Which integration technique should I use for something similar to:
(1 + x^2)^(1/2) dx
Trig substitution, I'd say.
No. All of the "trig substutions" are based on sin^2+ cos^2= 1 and so involve either "1-" or "-1".
Oh, now that was a silly thing for me to say! sin^2(x)+ cos^2(x)= 1 so, dividing through by cos^2(x), we have
tan^2(x)+ 1= sec^2(x), just as Sleek suggested. (I am delighted that, while pointing out that I was completely wrong, he referred to an earlier post of mine!)
Looks to me like a hyperbolic substition should work. Since cosh^2(y)- sinh^2(y)= 1, cosh^2(y)= 1+ sinh^2(y). Let x= sinh(y).
Even x=tan(m) would work. One would end up with int([sec(m)]^3) dm. This can be integrated using Int By Parts by differentiating sec(x) and integrating sec^2(x). Also, it can be integrated by method HallsofIvy suggested here: https://www.physicsforums.com/archive/index.php/t-156162.html.
The only down side is that sec^3 is usually quite a labororus integral to calculate :( Hall's hperbolic suggestion is the quickest.
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