Question Number 179513 by cortano1 last updated on 30/Oct/22
$$\:\mathrm{F}\left(\mathrm{x}\right)\:=\:\int\:\frac{\mathrm{1}}{\mathrm{x}}\:\sqrt{\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}}\:\mathrm{dx}\: \\ $$$$\:\:\mathrm{F}\left(\mathrm{1}\right)=\mathrm{0}\: \\ $$$$\:\:\mathrm{F}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=? \\ $$
Answered by mr W last updated on 30/Oct/22
$${let}\:{t}=\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}} \\ $$$${x}=\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$${dx}=\left[\frac{−\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }−\frac{\mathrm{2}{t}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\right]{dt}=\frac{−\mathrm{4}{t}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$${F}\left({x}\right)=\int\left(\frac{\mathrm{1}+{t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} }\right){t}\frac{−\mathrm{4}{t}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$\:\:\:=\int\frac{−\mathrm{4}{t}^{\mathrm{2}} }{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}{dt} \\ $$$$\:\:\:=\mathrm{2}\int\left(\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{1}−{t}^{\mathrm{2}} }\right){dt} \\ $$$$\:\:\:=\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} {t}−\mathrm{ln}\:\frac{\mathrm{1}+{t}}{\mathrm{1}−{t}}+{C} \\ $$$$\:\:\:=\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}−\mathrm{ln}\:\frac{\sqrt{\mathrm{1}+{x}}+\sqrt{\mathrm{1}−{x}}}{\:\sqrt{\mathrm{1}+{x}}−\sqrt{\mathrm{1}−{x}}}+{C} \\ $$$$\:{F}\left(\mathrm{1}\right)=\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{0}}{\mathrm{2}}}−\mathrm{ln}\:\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}}}+{C}=\mathrm{0} \\ $$$$\:\Rightarrow{C}=\mathrm{0} \\ $$$${F}\left({x}\right)=\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}−\mathrm{ln}\:\frac{\sqrt{\mathrm{1}+{x}}+\sqrt{\mathrm{1}−{x}}}{\:\sqrt{\mathrm{1}+{x}}−\sqrt{\mathrm{1}−{x}}} \\ $$$${F}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}−\mathrm{ln}\:\frac{\sqrt{\mathrm{3}}+\mathrm{1}}{\:\sqrt{\mathrm{3}}−\mathrm{1}} \\ $$$${F}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\pi}{\mathrm{3}}−\mathrm{ln}\:\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)\:=\frac{\pi}{\mathrm{3}}+\mathrm{ln}\:\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)\:\checkmark \\ $$
Commented by cortano1 last updated on 30/Oct/22
$$\:\mathrm{i}\:\mathrm{got}\:\mathrm{F}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\pi}{\mathrm{3}}+\mathrm{ln}\:\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)\: \\ $$
Commented by Tawa11 last updated on 30/Oct/22
$$\mathrm{Great}\:\mathrm{sir}. \\ $$
Commented by mr W last updated on 30/Oct/22
$${thanks}\:{for}\:{understanding}! \\ $$
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
$${i}'{ll}\:{ask}\:{tinku}\:{tara}\:{to}\:{add}\:{the}\:{feature} \\ $$$${that}\:{one}\:{receives}\:{notifications}\:{also} \\ $$$${when}\:{his}\:{bookmarked}\:{posts}\:{get}\: \\ $$$${updated}. \\ $$
Commented by Tawa11 last updated on 30/Oct/22
$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{Mr}\:\mathrm{brandon}\:\mathrm{got}\:\mathrm{it}. \\ $$$$\mathrm{I}\:\mathrm{like}\:\mathrm{to}\:\mathrm{save}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{for}\:\mathrm{future}\:\mathrm{purpose}. \\ $$
Commented by Tawa11 last updated on 30/Oct/22
$$\mathrm{Read}\:\mathrm{sir},\:\mathrm{I}\:\mathrm{get}\:\mathrm{your}\:\mathrm{point}\:\mathrm{sir}. \\ $$
Commented by mr W last updated on 30/Oct/22
$${i}\:{understood}\:{that}\:{you}\:{just}\:{want}\:{to}\: \\ $$$${track}\:{the}\:{posts}\:{in}\:{this}\:{way},\:{instead}\:{of} \\ $$$${really}\:{commenting}\:{the}\:{posts}. \\ $$$${bookmarking}\:{the}\:{posts}\:{is}\:{a}\:{better}\:{way}. \\ $$$${but}\:{you}\:{won}'{t}\:{be}\:{notificated}\:{when}\:{the} \\ $$$${posts}\:{are}\:{updated}.\:{therefore}\:{i}\:{want}\:{to} \\ $$$${asl}\:{tinku}\:{tara}\:{adding}\:{a}\:{new}\:{feature}, \\ $$$${that}\:{one}\:{gets}\:{notifications}\:{when}\:{the} \\ $$$${bookmarked}\:{posted}\:{get}\:{updated}. \\ $$
Commented by Rasheed.Sindhi last updated on 30/Oct/22
$$\mathrm{And}\:\mathrm{we}\:\mathrm{be}\:\mathrm{happy}\:\mathrm{that}\:\mathrm{miss}\:\mathrm{tawa}\:\mathrm{has} \\ $$$$\mathrm{admired}\:\mathrm{us}!!!\:{hahaha}…\: \\ $$
Commented by Ar Brandon last updated on 30/Oct/22
Commented by Tawa11 last updated on 31/Oct/22
$$\mathrm{Sir},\:\mathrm{tagging}\:\mathrm{your}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{save}\:\mathrm{it}. \\ $$$$\mathrm{Tagging}\:\mathrm{not}\:\mathrm{to}\:\mathrm{loose}\:\mathrm{your}\:\mathrm{solutions}. \\ $$$$\mathrm{Trying}\:\mathrm{my}\:\mathrm{own}\:\mathrm{way}\:\mathrm{to}\:\mathrm{save}\:\mathrm{your}\:\mathrm{solutions}\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{evidence}\:\mathrm{that}\:\mathrm{I}\:\mathrm{admire}\:\mathrm{your}\:\mathrm{solutions}. \\ $$