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Question Number 85480 by M±th+et£s last updated on 22/Mar/20

∫((csc(x))/(cos(x)+cos^3 (x)+...+cos^(2n+1) (x)))dx ∀x∈n

$$\int\frac{{csc}\left({x}\right)}{{cos}\left({x}\right)+{cos}^{\mathrm{3}} \left({x}\right)+...+{cos}^{\mathrm{2}{n}+\mathrm{1}} \left({x}\right)}{dx} \\ $$$$\forall{x}\in{n} \\ $$

Answered by mind is power last updated on 22/Mar/20

∫((csc(x))/(cos(x)(1+cos^2 (x)+.......+cos^(2n) (x))))dx =∫((csc(x)(1−cos^2 (x))dx)/(cos(x)(1−cos^(2n+2) (x)))) =∫((sin(x))/(cos(x)(1−cos^(2n+2) (x))))dx t=cos(x) =∫((−dt)/(t(1−t^(2n+2) )))=∫(dt/(t(t^(2n+2) −1))) =∫(dt/(t^(2n+3) (1−(1/t^(2n+2) )))) u=(1/t^(2n+2) )⇒du=((−2n−2)/t^(2n+3) )dt ⇔∫(du/((−2n−2)(1−u))) =((ln(u−1))/(2n+2))+c =((ln((1/(cos^(2n+2) ))−1))/(2n+2))+c

$$\int\frac{{csc}\left({x}\right)}{{cos}\left({x}\right)\left(\mathrm{1}+{cos}^{\mathrm{2}} \left({x}\right)+.......+{cos}^{\mathrm{2}{n}} \left({x}\right)\right)}{dx} \\ $$$$=\int\frac{{csc}\left({x}\right)\left(\mathrm{1}−{cos}^{\mathrm{2}} \left({x}\right)\right){dx}}{{cos}\left({x}\right)\left(\mathrm{1}−{cos}^{\mathrm{2}{n}+\mathrm{2}} \left({x}\right)\right)} \\ $$$$=\int\frac{{sin}\left({x}\right)}{{cos}\left({x}\right)\left(\mathrm{1}−{cos}^{\mathrm{2}{n}+\mathrm{2}} \left({x}\right)\right)}{dx} \\ $$$${t}={cos}\left({x}\right) \\ $$$$=\int\frac{−{dt}}{{t}\left(\mathrm{1}−{t}^{\mathrm{2}{n}+\mathrm{2}} \right)}=\int\frac{{dt}}{{t}\left({t}^{\mathrm{2}{n}+\mathrm{2}} −\mathrm{1}\right)} \\ $$$$=\int\frac{{dt}}{{t}^{\mathrm{2}{n}+\mathrm{3}} \left(\mathrm{1}−\frac{\mathrm{1}}{{t}^{\mathrm{2}{n}+\mathrm{2}} }\right)} \\ $$$${u}=\frac{\mathrm{1}}{{t}^{\mathrm{2}{n}+\mathrm{2}} }\Rightarrow{du}=\frac{−\mathrm{2}{n}−\mathrm{2}}{{t}^{\mathrm{2}{n}+\mathrm{3}} }{dt} \\ $$$$\Leftrightarrow\int\frac{{du}}{\left(−\mathrm{2}{n}−\mathrm{2}\right)\left(\mathrm{1}−{u}\right)} \\ $$$$=\frac{{ln}\left({u}−\mathrm{1}\right)}{\mathrm{2}{n}+\mathrm{2}}+{c} \\ $$$$=\frac{{ln}\left(\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{2}} }−\mathrm{1}\right)}{\mathrm{2}{n}+\mathrm{2}}+{c} \\ $$$$ \\ $$

Commented by M±th+et£s last updated on 22/Mar/20

great sir thank you so much

$${great}\:{sir}\:{thank}\:{you}\:{so}\:{much} \\ $$

Commented by mind is power last updated on 22/Mar/20

withe pleasur

$${withe}\:{pleasur} \\ $$

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