Question Number 183243 by cortano1 last updated on 24/Dec/22
$$\:\:\int\:\frac{\sqrt{\mathrm{sin}\:{x}}\:−\sqrt{\mathrm{cos}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}\:+\:\sqrt{\mathrm{cos}\:{x}}}\:{dx}\:=? \\ $$
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](Q183286.png)
$$\int\frac{\sqrt{\mathrm{sin}\:{x}}−\sqrt{\mathrm{cos}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}+\sqrt{\mathrm{cos}\:{x}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{tan}\:{x}}\:\rightarrow\:{dx}=\frac{\mathrm{2}{t}}{{t}^{\mathrm{4}} +\mathrm{1}}{dt}\right] \\ $$$$=\mathrm{2}\int\frac{{t}\left({t}−\mathrm{1}\right)}{\left({t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} −\sqrt{\mathrm{2}}{t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} +\sqrt{\mathrm{2}}{t}+\mathrm{1}\right)}{dt} \\ $$$$\mathrm{now}\:\mathrm{decompose}\:\mathrm{and}\:\mathrm{solve} \\ $$