Question Number 180545 by MathsFan last updated on 13/Nov/22
$$\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{for}} \\ $$$$\:\int\boldsymbol{\mathrm{cosec}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{and}} \\ $$$$\:\int\boldsymbol{\mathrm{sec}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{tan}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}} \\ $$$$\: \\ $$
Commented by Frix last updated on 13/Nov/22
$$\mathrm{use}\:{t}=\mathrm{tan}\:{x}\:\mathrm{with}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{and}\:{t}=\mathrm{cos}\:{x}\:\mathrm{with}\:\mathrm{the}\:\mathrm{2}^{\mathrm{nd}} \\ $$
Commented by MathsFan last updated on 13/Nov/22
$$\mathrm{thanks} \\ $$
Commented by Acem last updated on 13/Nov/22
$${Prove}\:{like}\:{what}? \\ $$$$\:{you}\:{mean}\:{prove}\:{that}\:\int\mathrm{sec}\:{x}\:\mathrm{tan}\:{x}\:{dx}\:=\:\mathrm{sec}\:{x}\:+{c}\:? \\ $$$${If}\:{that}\:{is}\:{waht}\:{you}\:{mean}\:{so}\:{suppose}\:{that} \\ $$$$\:\mathrm{sec}\:{x}=\:{t}\:\Rightarrow\:\mathrm{sec}\:{x}\:\mathrm{tan}\:{x}\:{dx}=\:{dt} \\ $$$$ \\ $$$${For}\:{the}\:\mathrm{1}{st}\:{suppose}\:{that}\:\mathrm{cot}\:{x}=\:{t}\:\Rightarrow\:−\mathrm{cosec}^{\mathrm{2}} \:{x}=\:{dt} \\ $$