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Question Number 79500 by ubaydulla last updated on 25/Jan/20

∫(cot^2 x+cot^4 x)dx

$$\int\left(\mathrm{cot}\:^{\mathrm{2}} {x}+\mathrm{cot}\:^{\mathrm{4}} {x}\right){dx} \\ $$

Answered by john santu last updated on 25/Jan/20

∫cot^2 x(1+cot^2 x)dx =  ∫cot^2 x cosec^2 xdx=  ∫−cot^2 x d(cot x) =  −(1/3)cot^3 x + c

$$\int\mathrm{cot}\:^{\mathrm{2}} {x}\left(\mathrm{1}+\mathrm{cot}\:^{\mathrm{2}} {x}\right){dx}\:= \\ $$$$\int\mathrm{cot}\:^{\mathrm{2}} {x}\:\mathrm{cosec}\:^{\mathrm{2}} {xdx}= \\ $$$$\int−\mathrm{cot}\:^{\mathrm{2}} {x}\:{d}\left(\mathrm{cot}\:{x}\right)\:= \\ $$$$−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{cot}\:^{\mathrm{3}} {x}\:+\:{c} \\ $$

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