F-x-1-x-1-x-1-x-dx-F-1-0-F-1-2-

Question Number 179513 by cortano1 last updated on 30/Oct/22

F (x) = \int \frac{1}{x} \sqrt{\frac{1 - x}{1 + x}} dx

F (1) = 0

F (\frac{1}{2}) = ?

Answered by mr W last updated on 30/Oct/22

let t=(√((1−x)/(1+x))) x=((1−t^2 )/(1+t^2 )) dx=[((−2t)/(1+t^2 ))−((2t(1−t^2 ))/((1+t^2 )^2 ))]dt=((−4t)/((1+t^2 )^2 ))dt F(x)=∫(((1+t^2 )/(1−t^2 )))t((−4t)/((1+t^2 )^2 ))dt =∫((−4t^2 )/((1+t^2 )(1−t^2 )))dt =2∫((1/(1+t^2 ))−(1/(1−t^2 )))dt =2 tan^(−1) t−ln ((1+t)/(1−t))+C =2 tan^(−1) (√((1−x)/(1+x)))−ln (((√(1+x))+(√(1−x)))/( (√(1+x))−(√(1−x))))+C F(1)=2 tan^(−1) (√(0/2))−ln ((√2)/( (√2)))+C=0 ⇒C=0 F(x)=2 tan^(−1) (√((1−x)/(1+x)))−ln (((√(1+x))+(√(1−x)))/( (√(1+x))−(√(1−x)))) F((1/2))=2 tan^(−1) (1/( (√3)))−ln (((√3)+1)/( (√3)−1)) F((1/2))=(π/3)−ln (2+(√3)) =(π/3)+ln (2−(√3)) ✓

l e t t = \sqrt{\frac{1 - x}{1 + x}}

x = \frac{1 - t^{2}}{1 + t^{2}}

d x = [\frac{- 2 t}{1 + t^{2}} - \frac{2 t (1 - t^{2})}{{(1 + t^{2})}^{2}}] d t = \frac{- 4 t}{{(1 + t^{2})}^{2}} d t

F (x) = \int (\frac{1 + t^{2}}{1 - t^{2}}) t \frac{- 4 t}{{(1 + t^{2})}^{2}} d t

= \int \frac{- 4 t^{2}}{(1 + t^{2}) (1 - t^{2})} d t

= 2 \int (\frac{1}{1 + t^{2}} - \frac{1}{1 - t^{2}}) d t

= 2 \tan^{- 1} t - \ln \frac{1 + t}{1 - t} + C

= 2 \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} - \ln \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} - \sqrt{1 - x}} + C

F (1) = 2 \tan^{- 1} \sqrt{\frac{0}{2}} - \ln \frac{\sqrt{2}}{\sqrt{2}} + C = 0

\Rightarrow C = 0

F (x) = 2 \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} - \ln \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} - \sqrt{1 - x}}

F (\frac{1}{2}) = 2 \tan^{- 1} \frac{1}{\sqrt{3}} - \ln \frac{\sqrt{3} + 1}{\sqrt{3} - 1}

F (\frac{1}{2}) = \frac{π}{3} - \ln (2 + \sqrt{3}) = \frac{π}{3} + \ln (2 - \sqrt{3}) ✓

Commented by cortano1 last updated on 30/Oct/22

i got F (\frac{1}{2}) = \frac{π}{3} + \ln (2 - \sqrt{3})

Commented by Tawa11 last updated on 30/Oct/22

Great sir .

Commented by mr W last updated on 30/Oct/22

t h a n k s f o r u n d e r s t a n d i n g!

Commented by ARUNG_Brandon_MBU last updated on 30/Oct/22

��The reason for the "Great Sir" on all posts according to me is for them to be saved in "My Post" section so they can easily be found �� Miss Tawa, instead of using "Great Sir" to save posts you can instead use the bookmark option�� unless it's not what I think.

Commented by mr W last updated on 30/Oct/22

i′ll ask tinku tara to add the feature that one receives notifications also when his bookmarked posts get updated.

i^{'} l l a s k t i n k u t a r a t o a d d t h e f e a t u r e

t h a t o n e r e c e i v e s n o t i f i c a t i o n s a l s o

w h e n h i s b o o k m a r k e d p o s t s g e t

u p d a t e d .

Commented by Tawa11 last updated on 30/Oct/22

Thanks sir. Mr brandon got it. I like to save the solutions for future purpose.

Thanks sir . Mr brandon got it .

I like to save the solutions for future purpose .

Commented by Tawa11 last updated on 30/Oct/22

Read sir, I get your point sir .

Commented by mr W last updated on 30/Oct/22

i understood that you just want to track the posts in this way, instead of really commenting the posts. bookmarking the posts is a better way. but you won′t be notificated when the posts are updated. therefore i want to asl tinku tara adding a new feature, that one gets notifications when the bookmarked posted get updated.

i u n d e r s t o o d t h a t y o u j u s t w a n t t o

t r a c k t h e p o s t s i n t h i s w a y, i n s t e a d o f

r e a l l y c o m m e n t i n g t h e p o s t s .

b o o k m a r k i n g t h e p o s t s i s a b e t t e r w a y .

b u t y o u {w o n}^{'} t b e n o t i f i c a t e d w h e n t h e

p o s t s a r e u p d a t e d . t h e r e f o r e i w a n t t o

a s l t i n k u t a r a a d d i n g a n e w f e a t u r e,

t h a t o n e g e t s n o t i f i c a t i o n s w h e n t h e

b o o k m a r k e d p o s t e d g e t u p d a t e d .

Commented by Rasheed.Sindhi last updated on 30/Oct/22

And we be happy that miss tawa has admired us!!! hahaha...

And we be happy that miss tawa has

admired us!!! h a h a h a \dots

Commented by Ar Brandon last updated on 30/Oct/22

��

Commented by Tawa11 last updated on 31/Oct/22

Sir, tagging your solutions to save it. Tagging not to loose your solutions. Trying my own way to save your solutions is an evidence that I admire your solutions.

Sir, tagging your solutions to save it .

Tagging not to loose your solutions .

Trying my own way to save your solutions is an

evidence that I admire your solutions .

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