You try a substitution to see whether it works.
The integrand names its own substitution. $\sqrt{a^2-x^2}$ asks for $x=a\sin\theta$; $a^2+x^2$ asks for $x=a\tan\theta$; an inner function whose derivative also appears asks for $u$ equal to it.
Next time: Name the object you want gone before you write $u$.