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

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

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

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

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

$$greatsirthankyousomuch$$

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

$$withepleasur$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com