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Question Number 119021 by A8;15: last updated on 21/Oct/20

Answered by MJS_new last updated on 21/Oct/20

$$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx=\int \frac{\sqrt{\tan x}}{1+\sqrt{\tan x}}dx=$$$$[t=\sqrt{\tan x}\to dx=\frac{2t}{t^4+1}dt]$$$$=2\int \frac{t^2}{(t+1)(t^4+1)}dt$$$$\mathrm{now}\mathrm{decompose}$$$$\mathrm{I}\mathrm{get}$$$$\ln (t+1)-\frac{1-\sqrt{2}}{4}\ln (t^2-\sqrt{2}t+1)-\frac{1+\sqrt{2}}{4}\ln (t^2+\sqrt{2}t+1)+$$$$+\frac{1}{2}(\arctan (\sqrt{2}t-1)-\arctan (\sqrt{2}t+1))+C$$

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