# $\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ (Integral of a trigonometric function under square root)

$$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$$ is an integral which I am not able to solve. I have restricted my ideas on trigonometric substitution but cannot conclude to an answer...will really appreciate if someone could help me out..or suggest an alternative method for the same ..

• $$\sqrt{1-\tan x}\mapsto \frac{\sqrt{1-x}}{1+x^2}\mapsto \frac{\sqrt{x}}{1+(1-x)^2}\mapsto \frac{x^2}{1+(1-x^2)^2}$$ and partial fraction decomposition. – Jack D'Aurizio Nov 2 '18 at 5:06

Substituting $$u=\sqrt{1-\tan x}$$ gives a pretty easy integral.

• Okay sir...will try this out and update soon.thank you. – user611339 Nov 2 '18 at 5:05