Задача 61604 ...
Условие
математика ВУЗ
157
Решение
★
x=1/sint
dx=-costdt/sin^2t
x^2-1=(1/sint)^2-1=(1-sin^2t)/sin^2t=cos^2t/sin^2t
sqrt(x^2-1)=cost/sint
sqrt((x^2-1)^7)=(cost/sint)^7
получаем
=∫(sin^7t)(-cost)dt/(cos^7t)(sin^2t)=- ∫ sin^5tdt/cos^6t=- ∫( sin^2t)^2*sint dt/(cos^6t)=
=-∫ (1-cos^2t)^2*sint dt/cos^6t=- ∫ (1-2cos^2t+cos^4t)sint dt/cos^6t=
=- ∫ (sintdt/cos^6t)+2 ∫ (sintdt)/cos^4t- ∫ (sintdt)/cos^2t=
=∫ d(cost)/cos^6t)-2 ∫ (dcost)/cos^4t+ ∫ (d(cost)/cos^2t=
=-1/(5cos^5t)+2/(3cos^3t)-(1/cost) + C