Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 136440 by liberty last updated on 22/Mar/21

∫ (dx/(sin^6 x)) ?

$$\:\int\:\frac{{dx}}{\mathrm{sin}\:^{\mathrm{6}} {x}}\:? \\ $$

Answered by rs4089 last updated on 22/Mar/21

∫cosec^6 x.dx ∫(1+cot^2 x)^2 cosec^2 x.dx let cotx=t ⇒ −cosec^2 x.dx=dt ∫(1+t^2 )^2 (−dt) −∫(1+t^4 +2t^2 )dt −t−(t^5 /5)−2(t^3 /3)+C −cotx−((cot^5 x)/5)−2((cot^3 x)/3)+C

$$\int{cosec}^{\mathrm{6}} {x}.{dx} \\ $$$$\int\left(\mathrm{1}+{cot}^{\mathrm{2}} {x}\right)^{\mathrm{2}} {cosec}^{\mathrm{2}} {x}.{dx} \\ $$$${let}\:\:{cotx}={t}\:\Rightarrow\:−{cosec}^{\mathrm{2}} {x}.{dx}={dt} \\ $$$$\int\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} \left(−{dt}\right) \\ $$$$−\int\left(\mathrm{1}+{t}^{\mathrm{4}} +\mathrm{2}{t}^{\mathrm{2}} \right){dt} \\ $$$$−{t}−\frac{{t}^{\mathrm{5}} }{\mathrm{5}}−\mathrm{2}\frac{{t}^{\mathrm{3}} }{\mathrm{3}}+{C} \\ $$$$−{cotx}−\frac{{cot}^{\mathrm{5}} {x}}{\mathrm{5}}−\mathrm{2}\frac{{cot}^{\mathrm{3}} {x}}{\mathrm{3}}+{C} \\ $$

Answered by EDWIN88 last updated on 22/Mar/21

E = ∫ (dx/(sin^6 x)) = ∫ (((sin^2 x+cos^2 x)^2 )/(sin^6 x)) dx E= ∫ ((sin^4 x+2sin^2 x cos^2 x + cos^4 x)/(sin^6 x)) dx E=∫ cosec^2 x dx+∫ cot^4 x cosec^2 x dx+2∫ cot^2 x cosec^2 x dx E = −cot x−∫ cot^4 x d(cot x)−2∫ cot^2 x d(cot x) E=−cot x−((cot^5 x)/5)−((2cot^3 x)/3) + c

$$\mathrm{E}\:=\:\int\:\frac{\mathrm{dx}}{\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}}\:=\:\int\:\frac{\left(\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}\right)^{\mathrm{2}} \:}{\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}}\:\mathrm{dx} \\ $$$$\mathrm{E}=\:\int\:\frac{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}+\mathrm{2sin}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:+\:\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}{\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}}\:\mathrm{dx} \\ $$$$\mathrm{E}=\int\:\mathrm{cosec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}+\int\:\mathrm{cot}\:^{\mathrm{4}} \mathrm{x}\:\mathrm{cosec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}+\mathrm{2}\int\:\mathrm{cot}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{cosec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx} \\ $$$$\mathrm{E}\:=\:−\mathrm{cot}\:\mathrm{x}−\int\:\mathrm{cot}\:^{\mathrm{4}} \mathrm{x}\:\mathrm{d}\left(\mathrm{cot}\:\mathrm{x}\right)−\mathrm{2}\int\:\mathrm{cot}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{d}\left(\mathrm{cot}\:\mathrm{x}\right) \\ $$$$\mathrm{E}=−\mathrm{cot}\:\mathrm{x}−\frac{\mathrm{cot}\:^{\mathrm{5}} \mathrm{x}}{\mathrm{5}}−\frac{\mathrm{2cot}\:^{\mathrm{3}} \mathrm{x}}{\mathrm{3}}\:+\:\mathrm{c}\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com