Question Number 186780 by normans last updated on 10/Feb/23
$$ \\ $$$$\:\:\:\:\:\:\:\int\:\:\:\frac{\mathrm{1}\:+\:\boldsymbol{{sin}}\:\boldsymbol{{x}}\:+\:\boldsymbol{{cos}}\:\boldsymbol{{x}}}{\mathrm{1}\:+\:\boldsymbol{{sin}}\:\boldsymbol{{x}}}\:\:\boldsymbol{{dx}}\:\: \\ $$$$ \\ $$
Answered by Ar Brandon last updated on 10/Feb/23
$${I}=\int\frac{\mathrm{1}+\mathrm{sin}{x}+\mathrm{cos}{x}}{\mathrm{1}+\mathrm{sin}{x}}{dx} \\ $$$$\:\:=\int\left(\mathrm{1}+\frac{\mathrm{cos}{x}}{\mathrm{1}+\mathrm{sin}{x}}\right){dx} \\ $$$$\:\:={x}+\mathrm{ln}\left(\mathrm{1}+\mathrm{sin}{x}\right)+{C} \\ $$