What is the most efficient integration technique for (1 + x^2)^(1/2) dx?

[arevolutionist]

Which integration technique should I use for something similar to:

(1 + x^2)^(1/2) dx

 

[TMM]

Trig substitution, I'd say.

 

[HallsofIvy]

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).

 

[Sleek]

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.

Regards,
Sleek.

 

[Gib Z]

The only down side is that sec^3 is usually quite a labororus integral to calculate :( Hall's hperbolic suggestion is the quickest.