How to integrate $\frac1{1+\sqrt{\tan x}}$?

Asked Nov 11 '23 at 16:48

Active Nov 11 '23 at 17:07

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How to calculate $$\int\frac1{1+\sqrt{\tan x}}dx?$$ The integrand is$$\frac1{1+\sqrt{\frac{\sin x}{\cos x}}}=\frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}}$$ $$=\frac12\frac{(\sqrt{\cos x}+\sqrt{\sin x})+(\sqrt{\cos x}-\sqrt{\sin x})}{\sqrt{\cos x}+\sqrt{\sin x}}$$ $$=\frac12\left(1+\frac{\sqrt{\cos x}-\sqrt{\sin x}}{\sqrt{\cos x}+\sqrt{\sin x}}\right).$$

Now I am facing problem to calculate $$\int\frac{\sqrt{\cos x}-\sqrt{\sin x}}{\sqrt{\cos x}+\sqrt{\sin x}}dx.$$ Please help me out.

edited Nov 11 '23 at 17:07

Anne Bauval

49,005

asked Nov 11 '23 at 16:48

george

Try $u(x) = \tan(x)$ and use the relation $\sec^2(x) = \tan^2(x) + 1$ to start. – Jimeens Nov 11 '23 at 17:05
$u = \tan x $ then partial fractions – pie Nov 11 '23 at 17:05
If you do $u=\tan x,$ there remains a square root. Better follow the link above. – Anne Bauval Nov 11 '23 at 17:09
Similar question is answered here https://math.stackexchange.com/questions/875014/evaluation-of-int-frac-sqrt-sin-x-sqrt-sin-x-sqrt-cos-xdx/4746225#4746225 – Maverick Nov 12 '23 at 02:16

How to integrate $\frac1{1+\sqrt{\tan x}}$?

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