Question Number 91074 by M±th+et+s last updated on 27/Apr/20
$${this}\:{trig}\:{integral}\:{has}\:{quite}\:{a}\:{few}\: \\ $$$${insights}\:{on}\:{trig}\:{integrals}\:{andd}\: \\ $$$${u}\:{subs}\:{as}\:{well}\:{as}\:{on}\:{the}\:{properties} \\ $$$${of}\:{logarithms}.\:{try}\:{it}\:{out}\:{it}'{s}\:{a}\:{nice} \\ $$$${one} \\ $$$$\int{tan}\left({x}\right){dx} \\ $$
Commented by mathmax by abdo last updated on 28/Apr/20
$$\int\:{tanxdx}\:=\int\:\frac{{sinx}}{{cosx}}{dx}\:=−\int\frac{{d}\left({cosx}\right)}{{cosx}}\:=−{ln}\mid{cosx}\mid\:+{C} \\ $$
Answered by MWSuSon last updated on 27/Apr/20
$${isn}'{t}\:{this}\:{just}\:−{log}_{{e}} \mathrm{cos}\:\left({x}\right)+{C}\:{or}\:{am}\:{i}\: \\ $$$${missing}\:{something}? \\ $$
Commented by M±th+et+s last updated on 28/Apr/20
$${there}\:{is}\:{another}\:{solutions}\:{like} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{tan}^{\mathrm{2}} \left({x}\right)\right)+{c} \\ $$$$ \\ $$
Commented by abdomathmax last updated on 28/Apr/20
$${yes}\:{because}\:\mathrm{1}+{tan}^{\mathrm{2}} {x}\:=\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}\rightarrow{the}\:{same}\:{result} \\ $$