Question Number 212141 by MrGaster last updated on 03/Oct/24
$$ \\ $$$$\:\:\:\:\:\:\:\:\int\frac{\mathrm{cos}^{\mathrm{2}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{dx}. \\ $$$$ \\ $$
Answered by BHOOPENDRA last updated on 03/Oct/24
$$\int\frac{\mathrm{cos}\:^{\mathrm{2}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:{dx}\:\overset{{t}={tan}\left(\frac{{x}}{\mathrm{2}}\right)} {=}\: \\ $$$$\int\:\frac{\left({t}−\mathrm{1}\right)^{\mathrm{2}} \left({t}+\mathrm{1}\right)^{\mathrm{2}} }{\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left({t}^{\mathrm{2}} −\mathrm{2}{t}−\mathrm{1}\right)}{dt} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{1}}{\left({t}^{\mathrm{2}} −\mathrm{2}{t}−\mathrm{1}\right)}{dt}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{1}}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)}{dt}+\int\frac{\left({t}−\mathrm{1}\right)}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\frac{\mathrm{ln}\:\left({t}−\sqrt{\mathrm{2}}+\mathrm{1}\right)}{\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:−\frac{\mathrm{ln}\:\left({t}+\sqrt{\mathrm{2}}−\mathrm{1}\right)}{\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} }+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}^{−\mathrm{1}} \left({t}\right)−\frac{{t}}{\mathrm{2}\left({t}^{\mathrm{2}} +\mathrm{1}\right)}−\frac{\mathrm{1}}{\mathrm{2}\left({t}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\frac{{x}}{\mathrm{2}}+\frac{\mathrm{ln}\:\left(\mathrm{tan}\left(\frac{{x}}{\mathrm{2}}\right)−\sqrt{\mathrm{2}}+\mathrm{1}\right)}{\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} }+\frac{\mathrm{ln}\:\left(\mathrm{tan}\:\left(\frac{{x}}{\mathrm{2}}\right)+\sqrt{\mathrm{2}}−\mathrm{1}\right)}{\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\:\:\:\:\:−\frac{\mathrm{tan}\:\left(\frac{{x}}{\mathrm{2}}\right)}{\mathrm{2}\left(\mathrm{tan}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{1}\right)}\:−\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{tan}^{\mathrm{2}} \:\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{1}\right)}+{C} \\ $$
Answered by Frix last updated on 03/Oct/24
$$\int\frac{\mathrm{cos}^{\mathrm{2}} \:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{dx}= \\ $$$$=\int\left(\frac{\mathrm{cos}\:{x}\:−\mathrm{sin}\:{x}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}\right.}\right){dx}= \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\mathrm{cos}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\:{dx}+\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\int\frac{{dx}}{\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)}= \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\:+\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\mathrm{ln}\:\mid\mathrm{tan}\:\left(\frac{{x}}{\mathrm{2}}+\frac{\pi}{\mathrm{8}}\right)\mid\:+{C} \\ $$