Question

найти интеграл dx/(sin^2x*cos^4x) (замена t=tgx)
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Answer 1

$int dfrac{dx}{sin^2xcdot cos^4x}=int dfrac{1}{sin^2x}cdot dfrac{1}{cos^2x}cdot dfrac{dx}{cos^2x}=int (1+ctg^2x)cdot (1+tg^2x)cdot dfrac{dx}{cos^2x}=\\\=int Big(1+dfrac{1}{tg^2x}Big)(1+tg^2x)cdot d(tgx)=Big[; t=tgx; Big]=int dfrac{(1+t^2)^2}{t^2}, dt=\\\=int dfrac{1+2t^2+t^4}{t^2}, dt=int Big(dfrac{1}{t^2}+2+t^2}Big), dt=-dfrac{1}{t}+2t+dfrac{t^3}{3}+C=\\\=-dfrac{1}{tgx}+2tgx+dfrac{tg^3x}{3}+C$