Question Number 46841 by maxmathsup by imad last updated on 01/Nov/18
$${calculate}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{dx}}{{cosx}\:{sinx}} \\ $$
Commented by maxmathsup by imad last updated on 01/Nov/18
![we have I = ∫_(π/4) ^(π/3) (dx/(cosxsinx)) =2 ∫_(π/4) ^(π/3) (dx/(sin(2x))) =_(2x=t) 2 ∫_(π/2) ^((2π)/3) (1/(sin(t))) (dt/2) =∫_(π/2) ^((2π)/3) (dt/(sint)) =_(tan((t/2))=u) ∫_(tan((π/4))) ^(tan((π/3))) (1/((2u)/(1+u^2 ))) ((2du)/(1+u^2 )) = ∫_1 ^(√3) (du/u) =[ln∣u∣]_1 ^(√3) =ln((√3)) =((ln(3))/2) .](https://www.tinkutara.com/question/Q46875.png)
$${we}\:{have}\:{I}\:=\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{dx}}{{cosxsinx}}\:=\mathrm{2}\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\frac{{dx}}{{sin}\left(\mathrm{2}{x}\right)}\:=_{\mathrm{2}{x}={t}} \:\:\:\mathrm{2}\:\int_{\frac{\pi}{\mathrm{2}}} ^{\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\:\frac{\mathrm{1}}{{sin}\left({t}\right)}\:\frac{{dt}}{\mathrm{2}} \\ $$$$=\int_{\frac{\pi}{\mathrm{2}}} ^{\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\frac{{dt}}{{sint}}\:=_{{tan}\left(\frac{{t}}{\mathrm{2}}\right)={u}} \:\:\:\:\int_{{tan}\left(\frac{\pi}{\mathrm{4}}\right)} ^{{tan}\left(\frac{\pi}{\mathrm{3}}\right)} \:\:\:\:\:\frac{\mathrm{1}}{\frac{\mathrm{2}{u}}{\mathrm{1}+{u}^{\mathrm{2}} }}\:\frac{\mathrm{2}{du}}{\mathrm{1}+{u}^{\mathrm{2}} }\:=\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\:\frac{{du}}{{u}}\:=\left[{ln}\mid{u}\mid\right]_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \\ $$$$={ln}\left(\sqrt{\mathrm{3}}\right)\:=\frac{{ln}\left(\mathrm{3}\right)}{\mathrm{2}}\:. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Nov/18
$$\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \frac{{dx}}{{cos}^{\mathrm{2}} {x}×{tanx}} \\ $$$$\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\frac{{d}\left({tanx}\right)}{{tanx}} \\ $$$$\mid{ln}\left({tanx}\right)\mid_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \\ $$$$={ln}\left(\sqrt{\mathrm{3}}\:\right)−{ln}\left(\mathrm{1}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\mathrm{3} \\ $$