Integral of y=sqrt.(x^2+a^2) or y^2=x^2+a^2

[3hlang]

how do you integrate this...
y=sqrt.(x^2+a^2)

would greatly appreciate any help. thanks

 

[Cyosis]

Easiest way to do this is to make the the substitution x=a \sinh u. Note that \cosh^2 u-\sinh^2 u=1. Try it out! If you're totally unfamiliar with hyperbolic functions use x=\tan u instead.

 

[3hlang]

would i be right in thinking that cosh(x)=cos(ix) and sinh(x)=-isin(ix) and therefore
cosh(x)+sinh(x)=e^x?

 

[Cyosis]

Yes that's correct, although I don't see how this is relevant to your problem.

 

[HallsofIvy]

Since sin^2(u)+ cos^2(u)= 1 leads to tan^2(u)+ 1= sec^2(u) (divide both sides by cos^2(u)), you could also make the substitution ax= tan(u).