Question Number 193512 by SaRahAli last updated on 15/Jun/23
Answered by Subhi last updated on 15/Jun/23
$$\int\frac{\mathrm{1}}{\frac{\mathrm{1}}{{cos}\left({x}\right)}.\frac{{sin}\left({x}\right)}{{cos}\left({x}\right)}}\:\Rrightarrow\:\int\frac{{cos}^{\mathrm{2}} \left({x}\right)}{{sin}\left({x}\right)}.{dx} \\ $$$$\int\frac{\mathrm{1}−{sin}^{\mathrm{2}} \left({x}\right)}{{sin}\left({x}\right)}.{dx}\:\:\looparrowright\:\int{csc}\left({x}\right)\:+\:\int−{sin}\left({x}\right) \\ $$$$\int{csc}\left({x}\right)\:+{cos}\left({x}\right) \\ $$$$\int{csc}\left({x}\right).\frac{{csc}\left({x}\right)+{cot}\left({x}\right)}{{csc}\left({x}\right)+{cot}\left({x}\right)}\:+\:{cos}\left({x}\right) \\ $$$$−\int\frac{−{csc}\left({x}\right){cot}\left({x}\right)−{csc}^{\mathrm{2}} \left({x}\right)}{{csc}\left({x}\right)+{cot}\left({x}\right)}\:+\:{cos}\left({x}\right)\:\Rrightarrow\:−{ln}\mid{csc}\left({x}\right)+{cot}\left({x}\right)\mid+{cos}\left({x}\right) \\ $$
Commented by MM42 last updated on 15/Jun/23
$$\circledast\:\int\:\frac{{dx}}{{sinx}}=\int\frac{\mathrm{1}+{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{2}{tan}\frac{{x}}{\mathrm{2}}}\:{dx}=\:{ln}\left({tan}\frac{{x}}{\mathrm{2}}\right)+{c} \\ $$