Menu Close

csc-x-cos-x-cos-3-x-cos-2n-1-x-dx-x-n-




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

Leave a Reply

Your email address will not be published. Required fields are marked *