Задача 26680 Все на картинке...
Условие
Решение
dx=dt/cos^2t
= ∫ (1/tg^2t)*(cost)*(dt/cos^2t)=
=∫dt/(tg^2t*cost)=∫costdt/sin^2t=
=∫d(sint)/sin^2t=∫ sin^(-2)td(sint)=
=(-1/sint) + C
tgt=x
1+tg^2t=1+x^2
1/cos^2t=x^2+1
cost=1/sqrt(x^2+1)
sin^2t=1-cos^2t=1-(1/(x^2+1))=x^2/(x^2+1)
sint=x/sqrt(x^2+1)
О т в е т. sqrt(x^2+1)/x + C
2) x=3sint
dx=3costdt
= ∫ sqrt(9-9sin^2t) * 3 costdt/(81sin^4t)=)1/9) ∫ cos^2t/sin^4tdt=
(1/9) ∫ ctg^2t*(-dctgt)=-(1/9)(ctg^3t/3)+C=
=(-1/27)ctg^3t +C
sint=x/3
cost=sqrt(1-sin^2t)=sqrt(1-(x/3)^2)=sqrt(9-x^2)/3
ctg t=cost/sint=sqrt(9-x^2)/x
О т в е т.(-1/27) (sqrt(9-x^2)/x)^3+ C
3) x=2/sint
dx=-2costdt/sin^2t
sqrt(x^2-4)=sqrt((2/sint)^2-4)=2sqrt((1-sin^2t)/sin^2t)=2ctgt
= ∫ (2ctgt)*(sin^4t/16) *(-2costdt/sin^2t)=
= -(1/4)∫ cos^2t*sin^tdt= (1/4)∫ cos^2td(cost)=(1/12)cos^3t+C
sint=2/x
cost=sqrt((1-(2/x)^2)=sqrt(x^2-4)/x
=(1/12)(sqrt(x^2-4)/x)^3+C