Question Number 66335 by mathmax by abdo last updated on 12/Aug/19
$${find}\:\:\int_{−\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{6}}} \:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$
Commented by Prithwish sen last updated on 13/Aug/19
![∫_(−(π/6)) ^(π/6) ((sec^2 x)/(1+tanx)) dx = [ln(1+tanx)]_((−π)/6) ^(π/6) = ln(((√3)+1)/( (√3)−1))](https://www.tinkutara.com/question/Q66358.png)
$$\int_{−\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{6}}} \frac{\mathrm{sec}^{\mathrm{2}} \mathrm{x}}{\mathrm{1}+\mathrm{tanx}}\:\mathrm{dx}\:=\:\left[\mathrm{ln}\left(\mathrm{1}+\mathrm{tanx}\right)\right]_{\frac{−\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{6}}} \:=\:\mathrm{ln}\frac{\sqrt{\mathrm{3}}+\mathrm{1}}{\:\sqrt{\mathrm{3}}−\mathrm{1}}\: \\ $$