Задача №1452

\[ \int\frac{dx}{\sqrt{\left(a^2+x^2\right)^3}} =\left[\begin{aligned}& x=|a|\tg{t}.\\& dx=\frac{|a|dt}{\cos^2{t}}.\end{aligned}\right] =\int\frac{|a|dt}{\cos^2{t}\cdot\frac{|a|^3}{\cos^3{t}}} =\frac{1}{a^2}\int\cos{t}dt=\\ =\frac{\sin{t}}{a^2}+C =\frac{\tg{t}}{a\sqrt{1+\tg^2{t}}}+C =\frac{x}{a^2\sqrt{1+\frac{x^2}{a^2}}}+C =\frac{x}{a^2\sqrt{a^2+x^2}}+C \]