x=4tgt ⇒16+ x^2=16+16tg^2t=16(1+tg^2t)=16/cos^2t
(16+ x^2)^(3/2)=(4/cost)^3
dx=4dt/cos^2t
dx/(16+ x^2)^(3/2)=(4dt/cos^2t)/(4/cost)^3=cost/16
получим
[m] ∫_{0}^{\frac{π}{4}\frac{1}{16}cost dt=\frac{1}{16}(sint)|_{0}^{\frac{π}{4}=\frac{1}{16}\cdot \frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{32}[/m]