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).