How can I integrate $\int \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx\;$?

Question

$$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos{x}}}dx$$ Does a closed form exist for this integral? It has withstood every integration technique I know and WA shows an error ("try the following.." etc)

Are you sure it hasn't been given the integration limits? – user0102 Apr 05 '20 at 18:00 — user0102, Apr 05 '20 at 18:00

score 1 · Answer 1 · edited Apr 05 '20 at 21:21

1

Dividing top and bottom by $\sqrt{\sin x}$ you have $$ \int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \ dx = \int \frac{1}{1 + \sqrt{\cot x}} \ dx $$

edited Apr 05 '20 at 21:21

amWhy

210,739

answered Apr 05 '20 at 18:02

gt6989b

54,930

How can I integrate $\int \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx\;$?

1 Answers1