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Differentiation Questions

Question Number 187458 by mnjuly1970 last updated on 17/Feb/23

$$\\$$ $$\\$$ $$\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:\:\mathrm{2}{sin}\left({x}\right)\:−\:{cos}\left({x}\right)}{{sin}\left({x}\right)\:+\:{cos}\:\left({x}\right)}\:{dx}\:=\:? \\$$ $$−−−− \\$$

Answered by Frix last updated on 17/Feb/23

$$\frac{\mathrm{2sin}\:{x}\:−\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}= \\$$ $$=\frac{\mathrm{3}\left(\mathrm{sin}\:{x}\:−\mathrm{cos}\:{x}\right)}{\mathrm{2}\left(\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}\right)}+\frac{\mathrm{1}}{\mathrm{2}}= \\$$ $$=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{3}\sqrt{\mathrm{2}}\mathrm{cos}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)}{\mathrm{2}\sqrt{\mathrm{2}}\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)} \\$$ $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{3}}{\mathrm{2}}\mathrm{cot}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\right){dx}= \\$$ $$=\left[\frac{{x}}{\mathrm{2}}−\frac{\mathrm{3ln}\:\mid\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\mid}{\mathrm{2}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} =\frac{\pi}{\mathrm{4}} \\$$

Commented bymnjuly1970 last updated on 17/Feb/23

$${thanks}\:{alot} \\$$

Answered by anurup last updated on 17/Feb/23

$$\Omega=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{2sin}\:{x}−\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{dx}\:−\left(\mathrm{1}\right) \\$$ $$=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{2sin}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right)−\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right)}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right)+\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right)}{dx} \\$$ $$=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{2cos}\:{x}−\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}+\mathrm{sin}\:{x}}{dx}\:−\left(\mathrm{2}\right) \\$$ $$\left(\mathrm{1}\right)+\left(\mathrm{2}\right)\:\mathrm{gives} \\$$ $$\mathrm{2}\Omega=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{dx}\:\Rightarrow\Omega=\frac{\pi}{\mathrm{4}} \\$$

Commented bymnjuly1970 last updated on 17/Feb/23

$${thank}\:{you}\:{so}\:{much} \\$$