Question Number 25209 by Mr eaay last updated on 06/Dec/17
$$\int{cosec}\sqrt{\mathrm{4}{x}^{\mathrm{2}} }{dx} \\ $$$$ \\ $$
Answered by prakash jain last updated on 06/Dec/17
$$=\int\mathrm{cosec}\:\mathrm{2}{xdx} \\ $$$$=\int\frac{\mathrm{1}}{\mathrm{2sin}\:{x}\mathrm{cos}\:{x}}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} {x}\left(\frac{\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}\right)}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{cosec}^{\mathrm{2}} {x}}{\mathrm{cot}\:{x}}{dx} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{cot}\:{x}\right)+{C} \\ $$