Question Number 169362 by Mastermind last updated on 29/Apr/22
$$\int{tan}\left(\mathrm{2}{x}+\mathrm{3}\right){dx} \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by peter frank last updated on 29/Apr/22
$$\mathrm{let}\:\mathrm{u}=\mathrm{2x}+\mathrm{3} \\ $$$$\mathrm{du}=\mathrm{2dx} \\ $$$$\int\mathrm{tan}\:\mathrm{u}.\frac{\mathrm{du}}{\mathrm{2}} \\ $$
Commented by Mastermind last updated on 29/Apr/22
$${thanks},\:{you}\:{did}\:{not}\:{complete}\:{it}\:{tho}. \\ $$
Answered by Mathspace last updated on 29/Apr/22
$$\int\:{tan}\left(\mathrm{2}{x}+\mathrm{3}\right){dx}=_{\mathrm{2}{x}+\mathrm{3}={t}} \int\:{tant}\:\frac{{dt}}{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\:\frac{{sint}}{{cost}}{dt}\:=−\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid{cost}\mid\:+{c} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}{ln}\mid{cos}\left(\mathrm{2}{x}+\mathrm{3}\right)\mid\:+{c} \\ $$