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0-1-ln-x-1-x-dx-

Question Number 77232 by naka3546 last updated on 04/Jan/20 $$\underset{\mathrm{0}} {\int}\overset{\mathrm{1}} {\:}\:\mathrm{ln}\:\left(\:\sqrt{{x}}\:+\:\sqrt{\mathrm{1}−{x}}\:\right)\:{dx}\:\:=\:\:? \\ $$ Answered by MJS last updated on 04/Jan/20 $$\int\mathrm{ln}\:\left(\sqrt{{x}}+\sqrt{\mathrm{1}−{x}}\right)\:{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{arcsin}\:\sqrt{{x}}\:\rightarrow\:{dx}=\mathrm{2}\sqrt{{x}\left(\mathrm{1}−{x}\right)}{dt}\right] \\…

Question-77229

Question Number 77229 by Maclaurin Stickker last updated on 04/Jan/20 Commented by Maclaurin Stickker last updated on 04/Jan/20 $${ABC}\:{is}\:{a}\:{triangle}.\:{Radius}\:{circumferences} \\ $$$${shown}\:{in}\:{the}\:{figure}\:{are}\:{drawn} \\ $$$${internally}\:{to}\:{the}\:{triangle}\:{tanging} \\ $$$${two}\:{of}\:{its}\:{sides}\:{and}\:{the}\:{incircle}.…

some-thoughts-on-this-forum-we-cannot-solve-the-unanswered-questions-much-greater-mathematicians-haven-t-been-able-to-solve-yet-thus-it-makes-absolutely-no-sense-to-post-them-here-also-it-makes-no-

Question Number 77208 by MJS last updated on 04/Jan/20 $$\mathrm{some}\:\mathrm{thoughts}\:\mathrm{on}\:\mathrm{this}\:\mathrm{forum}… \\ $$$$\mathrm{we}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{unanswered}\:\mathrm{questions} \\ $$$$\mathrm{much}\:\mathrm{greater}\:\mathrm{mathematicians}\:\mathrm{haven}'\mathrm{t}\:\mathrm{been} \\ $$$$\mathrm{able}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{yet} \\ $$$$\mathrm{thus}\:\mathrm{it}\:\mathrm{makes}\:\mathrm{absolutely}\:\mathrm{no}\:\mathrm{sense}\:\mathrm{to}\:\mathrm{post} \\ $$$$\mathrm{them}\:\mathrm{here} \\ $$$$\mathrm{also}\:\mathrm{it}\:\mathrm{makes}\:\mathrm{no}\:\mathrm{sense}\:\mathrm{to}\:\mathrm{post}\:\mathrm{problems}\:\mathrm{from} \\ $$$$\mathrm{books}\:\mathrm{or}\:\mathrm{from}\:\mathrm{the}\:\mathrm{www}\:\mathrm{if}\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{have}\:\mathrm{the} \\…

x-x-30-2-1-2-

Question Number 142730 by Gbenga last updated on 04/Jun/21 $${x}^{{x}^{\mathrm{30}} } =\sqrt{}\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$ Commented by mr W last updated on 04/Jun/21 $${x}=\sqrt[{\mathrm{30}}]{\frac{\mathrm{15ln}\:\mathrm{2}}{\mathrm{2}{W}\left(\frac{\mathrm{15ln}\:\mathrm{2}}{\mathrm{2}}\right)}}\approx\mathrm{1}.\mathrm{045993524} \\ $$…

0-pi-2-sin-2t-1-xsin-2t-dt-

Question Number 142724 by lapache last updated on 04/Jun/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}\left(\mathrm{2}{t}\right)}{\mathrm{1}+{xsin}\left(\mathrm{2}{t}\right)}{dt}=…. \\ $$ Answered by Ar Brandon last updated on 04/Jun/21 $$\mathrm{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin2t}}{\mathrm{1}+\mathrm{xsin2t}}\mathrm{dt}=\frac{\mathrm{1}}{\mathrm{x}}\int_{\mathrm{0}}…