Integration Questions

Question Number 53295 by gunawan last updated on 20/Jan/19

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\mathrm{2}+\mathrm{cos}\:{x}}\:{dx}=... \\$$

Commented by maxmathsup by imad last updated on 20/Jan/19

$${changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}\:{give}\: \\$$$${A}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{2}+{cosx}}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} }\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{2}{dt}}{\mathrm{2}+\mathrm{2}{t}^{\mathrm{2}} +\mathrm{1}−{t}^{\mathrm{2}} }\:=\mathrm{2}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{3}+{t}^{\mathrm{2}} } \\$$$$=_{{t}\:=\sqrt{\mathrm{3}}{u}} \:\:\:\mathrm{2}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}} \:\:\frac{\sqrt{\mathrm{3}}{du}}{\mathrm{3}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}\:=\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}} \:\:\frac{{du}}{\mathrm{1}+{u}^{\mathrm{2}} }\:=\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}\:{arctan}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\right)=\frac{\mathrm{2}}{\sqrt{\mathrm{3}}}\:\frac{\pi}{\mathrm{6}} \\$$$$=\frac{\pi}{\mathrm{3}\sqrt{\mathrm{3}}}\:\Rightarrow\bigstar{I}\:=\frac{\pi}{\mathrm{3}\sqrt{\mathrm{3}}}\:\bigstar \\$$$$\\$$

Answered by tanmay.chaudhury50@gmail.com last updated on 20/Jan/19

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}−{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{1}+{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}}{dx} \\$$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{2}+\mathrm{2}{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}\:}+\mathrm{1}−{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx} \\$$$${t}={tan}\frac{{x}}{\mathrm{2}}\:\:\:\:{dt}=\frac{\mathrm{1}}{\mathrm{2}}{sec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}{dx} \\$$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{2}{dt}}{\mathrm{3}+{t}^{\mathrm{2}} } \\$$$$\mathrm{2}×\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\mid{tan}^{−\mathrm{1}} \left(\frac{{t}}{\sqrt{\mathrm{3}}}\right)\mid_{\mathrm{0}} ^{\mathrm{1}} \\$$$$\frac{\mathrm{2}}{\sqrt{\mathrm{3}}}\left[{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\right)\right]=\frac{\mathrm{2}}{\sqrt{\mathrm{3}}}×\frac{\pi}{\mathrm{6}}=\frac{\pi}{\mathrm{3}\sqrt{\mathrm{3}}} \\$$

Commented by gunawan last updated on 20/Jan/19

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{Sir} \\$$