[m]dx=\sqrt{5}costdt[/m]
[m]5-x^2=5-(\sqrt{5}sint)^2=5(1-sin^2t)=5cos^2t[/m]
[m](5-x^2)^{\frac{3}{2}}=(5cos^2t)^{\frac{3}{2}}=5^{\frac{3}{2}}cos^3t[/m]
[m] ∫ \frac{1}{(1-x^2)^{\frac{3}{2}}}dx= ∫ \frac{\sqrt{5}costdt}{5^{\frac{3}{2}}cos^3t}=\frac{1}{5} ∫ \frac{dt}{cos^2t}=\frac{1}{5}tgt+C[/m]
Обратный переход от переменной t к переменной х:
[m]x=\sqrt{5}sint[/m] ⇒ [m]sint=\frac{x}{\sqrt{5}}[/m]
[m]cost=\sqrt{1-(\frac{x}{\sqrt{5}})^2}[/m]
[m]tgt=\frac{sint}{cost}=\frac{\frac{x}{\sqrt{5}}}{\sqrt{1-(\frac{x}{\sqrt{5}})^2}}=\frac{x}{\sqrt{5-x^2}}[/m]
[m] ∫ \frac{1}{(1-x^2)^{\frac{3}{2}}}dx=\frac{1}{5}tgt+C=\frac{x}{5\sqrt{5-x^2}}+С[/m]