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sin-x-1-sin-x-cos-x-1-cos-x-dx-




Question Number 183976 by cortano1 last updated on 01/Jan/23
  ∫ ((sin x−(√(1+sin x)))/(cos x−(√(1+cos x)))) dx =?
$$\:\:\int\:\frac{\mathrm{sin}\:{x}−\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}{\mathrm{cos}\:{x}−\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}\:{dx}\:=? \\ $$
Answered by MJS_new last updated on 01/Jan/23
t=((1+sin (x/2))/(cos (x/2)))⇔x=2arctan ((t^2 −1)/(2t)) → dx=((4dt)/(t^2 −1))  ⇒  ∫((sin x −(√(1+sin x)))/(cos x −(√(1+cos x))))dx=  =4∫((t^4 −2t^3 +6t−1)/((t^2 +1)(t^2 +(√2)(1+(√5))t+1)(t^2 +(√2)(1−(√5))t+1)))dt  now do the decomposition etc. for yourself  because I′m too lazy again.
$${t}=\frac{\mathrm{1}+\mathrm{sin}\:\frac{{x}}{\mathrm{2}}}{\mathrm{cos}\:\frac{{x}}{\mathrm{2}}}\Leftrightarrow{x}=\mathrm{2arctan}\:\frac{{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{t}}\:\rightarrow\:{dx}=\frac{\mathrm{4}{dt}}{{t}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\Rightarrow \\ $$$$\int\frac{\mathrm{sin}\:{x}\:−\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}{\mathrm{cos}\:{x}\:−\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}{dx}= \\ $$$$=\mathrm{4}\int\frac{{t}^{\mathrm{4}} −\mathrm{2}{t}^{\mathrm{3}} +\mathrm{6}{t}−\mathrm{1}}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)\left({t}^{\mathrm{2}} +\sqrt{\mathrm{2}}\left(\mathrm{1}+\sqrt{\mathrm{5}}\right){t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} +\sqrt{\mathrm{2}}\left(\mathrm{1}−\sqrt{\mathrm{5}}\right){t}+\mathrm{1}\right)}{dt} \\ $$$$\mathrm{now}\:\mathrm{do}\:\mathrm{the}\:\mathrm{decomposition}\:\mathrm{etc}.\:\mathrm{for}\:\mathrm{yourself} \\ $$$$\mathrm{because}\:\mathrm{I}'\mathrm{m}\:\mathrm{too}\:\mathrm{lazy}\:\mathrm{again}. \\ $$

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