Задача 26650 ...

vk161478389

22.04.2018

интеграл 1/х^ 3√х^2-1 dx

SOVA

22.04.2018

x=1/sint
dx=-costdt/sin^2t

= ∫ (sin^3t*sint/cost)*(-costdt/sin^2t)=

= -∫ sin^2tdt=- ∫ (1-cos2t)/2=(-1/2)*t+(1/2)sin2t+C

sint=1/x ⇒ t=arcsin(1/x)
cost=sqrt(1-(1/x)^2)
cost=sqrt(x^2-1)/x


(-1/2)*t+(1/2)sin2t+C=(-1/2)*t+(1/2)*2sintcost+C
=(-1/2)*arcsin(1/x)+(1/x)*sqrt((x^2-1)/x) + C=

=(-1/2)*arcsin(1/x)+sqrt((x^2-1)/x^2) + C