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Question Number 183243 by cortano1 last updated on 24/Dec/22

  ∫ (((√(sin x)) −(√(cos x)))/( (√(sin x)) + (√(cos x)))) dx =?

$$\:\:\int\:\frac{\sqrt{\mathrm{sin}\:{x}}\:−\sqrt{\mathrm{cos}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}\:+\:\sqrt{\mathrm{cos}\:{x}}}\:{dx}\:=? \\ $$

Answered by MJS_new last updated on 24/Dec/22

∫(((√(sin x))−(√(cos x)))/( (√(sin x))+(√(cos x))))dx=       [t=(√(tan x)) → dx=((2t)/(t^4 +1))dt]  =2∫((t(t−1))/((t+1)(t^2 −(√2)t+1)(t^2 +(√2)t+1)))dt  now decompose and solve

$$\int\frac{\sqrt{\mathrm{sin}\:{x}}−\sqrt{\mathrm{cos}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}+\sqrt{\mathrm{cos}\:{x}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{tan}\:{x}}\:\rightarrow\:{dx}=\frac{\mathrm{2}{t}}{{t}^{\mathrm{4}} +\mathrm{1}}{dt}\right] \\ $$$$=\mathrm{2}\int\frac{{t}\left({t}−\mathrm{1}\right)}{\left({t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} +\sqrt{\mathrm{2}}{t}+\mathrm{1}\right)}{dt} \\ $$$$\mathrm{now}\:\mathrm{decompose}\:\mathrm{and}\:\mathrm{solve} \\ $$

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