Задача 15460 1/x^2*(x^2-4)dx...
Условие
предмет не задан
626
Решение
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х=2/cost
dx=2sintdt/cos^2t
∫dx/(x^2*(x^2-4))=
=∫2sintdt/(cos^2t*(4/cos^4t)*((4/cos^2t)-4)))=
=(1/8)∫cos^4tdt/sint=(1/8)∫(1-sin^2t)^2dt/sint=
=(1/8)∫(1-2sin^2t+sin^4t)dt/sint=
=(1/8)∫dt/sint -(1/4)∫sintdt +(1/8)∫sin^3tdt=
=(1/8) ln|tg(t/2)|-(1/4)*(-cost)-(1/8)∫(1-cos^2t)d(cost)=
=(1/8) ln|tg(t/2)|-(1/4)*(-cost)-(1/8)cost -(1/8)*(cos^3t/3)+C.
x=2/cost ⇒cost=2/x