∫(√(tan (x)))dx Who can solve this problem?


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Question Number 93304 by Shakhzod last updated on 12/May/20

$\int \sqrt{\tan (x)} d x W h o c a n s o l v e t h i s p r o b l e m ?$
Commented by john santu last updated on 12/May/20

$= - \frac{1}{\sqrt{2}} \tanh^{- 1} (\frac{\sqrt{\tan x} + \sqrt{\cot x}}{\sqrt{2}}) +$ $\frac{1}{\sqrt{2}} \tan^{- 1} (\frac{\sqrt{\tan x} - \sqrt{\cot x}}{\sqrt{2}}) + c$
Commented by i jagooll last updated on 12/May/20
cooll man ��
Commented by i jagooll last updated on 12/May/20

$set u = \sqrt{\tan x}, dx = \frac{2 u du}{u^{4} + 1}$ $\int \sqrt{\tan x} dx = \int \frac{2}{u^{2} + \frac{1}{u^{2}}} du$ $= \int \frac{1 - \frac{1}{u}}{{(u + \frac{1}{u})}^{2} - 2} du + \int \frac{1 + \frac{1}{u^{2}}}{{(u - \frac{1}{u})}^{2} + 2} du$ $[set u + \frac{1}{u} = \tan q], [w = u - \frac{1}{u}]$ $= \int \frac{1}{q^{2} - 2} dq + \int \frac{1}{w^{2} + 2} dw$ $= - \frac{1}{\sqrt{2}} \tanh^{- 1} (\frac{q}{\sqrt{2}}) + \frac{1}{\sqrt{2}} \tan^{- 1} (\frac{w}{\sqrt{2}}) + c$ $= - \frac{1}{\sqrt{2}} \tanh^{- 1} (\frac{\sqrt{\tan x} + \sqrt{\cot x}}{\sqrt{2}})$ $+ \frac{1}{\sqrt{2}} \tan^{- 1} (\frac{\sqrt{\tan x} - \sqrt{\cot x}}{\sqrt{2}}) + c$
Commented by M±th+et+s last updated on 12/May/20

$a t e n d p l s c h e c k f o r t y p o s a n d t h a n k y o u$
Commented by mathmax by abdo last updated on 12/May/20

$t h i s i n t e g r a l i s s o l v e d s e e t h e p l a t f o r m .$
	Answered by prakash jain last updated on 12/May/20

	Commented by Shakhzod last updated on 12/May/20

	$G r e a t . T h a n k y o u s o m u c h .$
	Answered by M±th+et+s last updated on 12/May/20

	$= \frac{1}{\sqrt{2}} \int \frac{2 s i n (x)}{\sqrt{s i n (2 x)}}$ $\frac{1}{\sqrt{2}} \int \frac{s i n (x) + c o s (x) + s i n (x) - c o s (x)}{\sqrt{s i n (2 x)}} . \frac{c o s (2 x)}{c o s (2 x)} d x$ $= \frac{1}{\sqrt{2}} \int \frac{1 + \sqrt{s i n (2 x)}}{\sqrt{1 - {s i n}^{2} (2 x)}} . \frac{c o s (2 x)}{\sqrt{s i n (2 x)}} d x + \frac{1}{\sqrt{2}} \int \frac{\sqrt{1 - s i n (2 x)}}{\sqrt{1 - {s i n}^{2} (2 x)}} . \frac{c o s (2 x)}{\sqrt{s i n (2 x)}} d x$ $= \frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{1 - {(\sqrt{s i n (2 x)})}^{2}}} . \frac{c o s (2 x)}{\sqrt{s i n (2 x)}} + \frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{1 + {(\sqrt{s i n (2 x)})}^{2}}} . \frac{c o s (2 x)}{\sqrt{s i n (2 x)}} d x$ $= \frac{1}{\sqrt{2}} {s i n}^{- 1} (\sqrt{s i n (2 x)}) + \frac{1}{\sqrt{2}} {s i n h}^{- 1} (\sqrt{s i n (2 x)}) + c$ $i c a n s o l v e i t w i t h u s i n g a n o t h e r w a y s$ $b u t i d o n t h a v e a t i m e n o w i w i l l p o s t$ $t h e m i n a n o t h e t i m e$
	Commented by Shakhzod last updated on 12/May/20

	$T h a n k y o u . I w i l l w a i t y o u t h e n .$
	Answered by M±th+et+s last updated on 12/May/20

	Answered by M±th+et+s last updated on 12/May/20

	Answered by M±th+et+s last updated on 12/May/20

	$= \frac{1}{\sqrt{2}} \int \frac{2 s i n (x)}{\sqrt{s i n (2 x)}} d x = \frac{1}{\sqrt{2}} \int \frac{c o s (x) + s i n (x) - c o s (x) + s i n (x)}{\sqrt{s i n (2 x)}} d x$ $= \frac{1}{\sqrt{2}} \int \frac{\sqrt{1 + s i n (2 x)} - \sqrt{1 - s i n (2 x)}}{\sqrt{s i n (2 x)}} . \frac{c s c (2 x) c o t (2 x)}{c s c (2 x) c o t (2 x)} d x$ $= \frac{1}{\sqrt{2}} \int \frac{(\sqrt{c s c (2 x) + 1} - \sqrt{c s c (2 x) - 1})}{\sqrt{{c s c}^{2} (2 x) - 1}} c o t (2 x) d x$ $\frac{1}{\sqrt{2}} \int \frac{c o t (2 x)}{\sqrt{c s c (2 x) - 1}} d x - \frac{1}{\sqrt{2}} \int \frac{c o t (2 x)}{\sqrt{c s c (2 x) + 1}} d x$ $\frac{1}{\sqrt{2}} \int \frac{\frac{c s c (2 x) c o t (2 x)}{\sqrt{c s c (2 x) - 1}}}{{(\sqrt{c s c 2 x - 1})}^{2} + 1} d x - \frac{1}{\sqrt{2}} \int \frac{\frac{c s c (2 x) c o t (2 x)}{\sqrt{c s c (2 x) + 1}}}{{(\sqrt{c s c (2 x) + 1})}^{2} - 1}$ $= - \frac{1}{\sqrt{2}} {t a n}^{- 1} (\sqrt{c s c (2 x) - 1}) - \frac{1}{\sqrt{2}} {t a n h}^{- 1} (\sqrt{c s c (2 x) + 1}) + c$
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