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Author: Tinku Tara

nice-calculus-prove-that-0-pi-4-xcot-x-dx-1-2-G-pi-4-log-2-

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Question Number 128245 by mnjuly1970 last updated on 05/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\:::\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {xcot}\left({x}\right){dx}=\frac{\mathrm{1}}{\mathrm{2}}\left({G}+\frac{\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)\right) \\ $$ Answered by Dwaipayan Shikari last updated on 05/Jan/21 $$\int_{\mathrm{0}}…

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1-just-bought-premium-I-would-be-willing-to-pay-more-I-suggest-a-donate-possibility-2-saving-an-email-doesn-t-work-at-least-the-website-returns-fail-

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Question Number 128241 by MJS_new last updated on 05/Jan/21 $$\left(\mathrm{1}\right)\:\mathrm{just}\:\mathrm{bought}\:\mathrm{premium}.\:\mathrm{I}\:\mathrm{would}\:\mathrm{be}\:\mathrm{willing} \\ $$$$\mathrm{to}\:\mathrm{pay}\:\mathrm{more}.\:\mathrm{I}\:\mathrm{suggest}\:\mathrm{a}\:“\mathrm{donate}''\:\mathrm{possibility} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{saving}\:\mathrm{an}\:\mathrm{email}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{work};\:\mathrm{at}\:\mathrm{least}\:\mathrm{the} \\ $$$$\mathrm{website}\:\mathrm{returns}\:“{fail}'' \\ $$ Commented by Tinku Tara last updated on…

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1-6-p-logp-p-2-1-log1-pi-2-log2-4pi-2-log3-9pi-2-p-prime-

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Question Number 128236 by Dwaipayan Shikari last updated on 05/Jan/21 $$\frac{\mathrm{1}}{\mathrm{6}}\underset{{p}} {\overset{\infty} {\sum}}\frac{{logp}}{{p}^{\mathrm{2}} −\mathrm{1}}=\frac{{log}\mathrm{1}}{\pi^{\mathrm{2}} }+\frac{{log}\mathrm{2}}{\mathrm{4}\pi^{\mathrm{2}} }+\frac{{log}\mathrm{3}}{\mathrm{9}\pi^{\mathrm{2}} }+…\:\:\:\left({p}={prime}\right) \\ $$ Commented by Dwaipayan Shikari last updated…

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Lines-5x-12y-10-0-and-5x-12y-40-0-touch-circle-C-1-of-diameter-6-If-the-center-of-C-1-lies-in-the-Ist-quadrant-find-the-equation-of-circle-C-2-which-is-concentric-with-C-1-and-cuts-intercep

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Question Number 62698 by Ankit0512 last updated on 24/Jun/19 $${Lines}\:\mathrm{5}{x}+\mathrm{12}{y}−\mathrm{10}=\mathrm{0}\:{and}\:\mathrm{5}{x}−\mathrm{12}{y}−\mathrm{40}=\mathrm{0} \\ $$$${touch}\:{circle}\:{C}_{\mathrm{1}} \:{of}\:{diameter}\:\mathrm{6}.\:{If}\:{the}\: \\ $$$${center}\:{of}\:{C}_{\mathrm{1}} \:{lies}\:{in}\:{the}\:{Ist}\:{quadrant}, \\ $$$${find}\:{the}\:{equation}\:{of}\:{circle}\:{C}_{\mathrm{2}} \:{which}\:{is}\: \\ $$$${concentric}\:{with}\:{C}_{\mathrm{1}\:} \:{and}\:{cuts}\:{intercept} \\ $$$${of}\:{length}\:\mathrm{8}\:{on}\:{these}\:{lines}. \\…

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1-For-n-N-define-s-n-1-2-2-n-2-2-n-1-Which-of-the-following-is-false-for-some-values-of-n-N-A-3-s-n-B-7-s-n-C-s-n-s-n-1-D-s-n-2-gt-s-n-1-E-12s-1-s-2-s-n-lt-s-n-1-

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Question Number 128232 by math35 last updated on 05/Jan/21 $$\mathrm{1}.\mathrm{For}\:\mathrm{n}\in\mathbb{N},\mathrm{define}\:\:\mathrm{s}_{\mathrm{n}} =\mathrm{1}+\mathrm{2}^{\mathrm{2}^{\mathrm{n}} } +\mathrm{2}^{\mathrm{2}^{\mathrm{n}+\mathrm{1}} } .\mathrm{Which}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{false}\:\mathrm{for}\:\mathrm{some}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\in\mathbb{N}? \\ $$$$\mathrm{A}.\mathrm{3}\mid\mathrm{s}_{\mathrm{n}} \\ $$$$\mathrm{B}.\mathrm{7}\mid\mathrm{s}_{\mathrm{n}} \\ $$$$\mathrm{C}.\mathrm{s}_{\mathrm{n}} \mid\mathrm{s}_{\mathrm{n}+\mathrm{1}} \\…

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nice-calculus-prove-that-0-pi-tan-1-tan-2-x-tan-2-x-dx-pi-2-pi-4-

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Question Number 128225 by mnjuly1970 last updated on 05/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\phi\:=\:\int_{\mathrm{0}} ^{\:\pi} \frac{{tan}^{−\mathrm{1}} \left({tan}^{\mathrm{2}} \left({x}\right)\right)}{{tan}^{\mathrm{2}} \left({x}\right.}\:{dx}\overset{???} {=}\pi\left(\sqrt{\mathrm{2}}\:−\frac{\pi}{\mathrm{4}}\right) \\ $$$$ \\ $$ Answered…

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