Question Number 151060 by qaz last updated on 18/Aug/21 $$\mathrm{Calculate}\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}\sqrt{\mathrm{x}}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\mathrm{ax}\right)}\mathrm{dx}=\frac{\mathrm{a}^{\mathrm{2}} −\mathrm{a}+\sqrt{\mathrm{2a}}}{\:\sqrt{\mathrm{2}}\mathrm{a}\left(\mathrm{1}+\mathrm{a}^{\mathrm{2}} \right)}\pi\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:,\left(\mathrm{a}>\mathrm{0}\right) \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 151063 by qaz last updated on 18/Aug/21 $$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\left(\pi^{\mathrm{2}} +\mathrm{ln}^{\mathrm{2}} \mathrm{x}\right)\mathrm{x}}\mathrm{dx}=\gamma \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 151062 by qaz last updated on 18/Aug/21 $$\mathrm{Calculate}\:\:::\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\centerdot\sqrt[{\mathrm{4}}]{\mathrm{8x}^{\mathrm{2}} +\mathrm{8x}+\mathrm{1}}}=\frac{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{8}}\right)}{\mathrm{2}^{\frac{\mathrm{11}}{\mathrm{4}}} \Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 85523 by TawaTawa1 last updated on 22/Mar/20 Commented by TawaTawa1 last updated on 22/Mar/20 $$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$ Answered by MJS last updated on…
Question Number 19986 by Tinkutara last updated on 19/Aug/17 $$\mathrm{An}\:\mathrm{aeroplane}\:\mathrm{has}\:\mathrm{to}\:\mathrm{go}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:{A} \\ $$$$\mathrm{to}\:\mathrm{point}\:{B},\:\mathrm{500}\:\mathrm{km}\:\mathrm{away}\:\mathrm{due}\:\mathrm{30}°\:\mathrm{east} \\ $$$$\mathrm{of}\:\mathrm{north}.\:\mathrm{A}\:\mathrm{wind}\:\mathrm{is}\:\mathrm{blowing}\:\mathrm{due}\:\mathrm{north} \\ $$$$\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{20}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{The}\:\mathrm{air}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{plane}\:\mathrm{is}\:\mathrm{150}\:\mathrm{ms}^{−\mathrm{1}} .\:\mathrm{Find}\:\mathrm{the}\:\mathrm{direction} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{pilot}\:\mathrm{should}\:\mathrm{head}\:\mathrm{the} \\ $$$$\mathrm{plane}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{point}\:{B}. \\…
Question Number 151059 by john_santu last updated on 18/Aug/21 $$\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{2}}\:+\mathrm{2}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{2}−\sqrt[{\mathrm{3}}]{\mathrm{x}+\mathrm{2}}}\:=\:\mathrm{2}\: \\ $$$$\:\:\:\:\mathrm{x}\:=? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 151053 by Tawa11 last updated on 17/Aug/21 Answered by puissant last updated on 18/Aug/21 $${Q}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{sin}\left({lnx}\right)−{ln}\left({x}\right)}{{ln}^{\mathrm{2}} {x}}{dx} \\ $$$${t}=−{lnx}\:\rightarrow\:{x}={e}^{−{t}} \rightarrow{dx}=−{e}^{−{t}} {dt} \\…
Question Number 151052 by mathdanisur last updated on 17/Aug/21 $$\mathrm{if}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c}\:\:\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\:\mathrm{and} \\ $$$$\frac{\mathrm{a}}{\mathrm{1}+\mathrm{a}}\:+\:\frac{\mathrm{b}}{\mathrm{1}+\mathrm{b}}\:+\:\frac{\mathrm{c}}{\mathrm{1}+\mathrm{c}}\:=\:\mathrm{1}\:\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{abc}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{8}} \\ $$ Answered by dumitrel last updated on 18/Aug/21 $$\mathrm{3}{r}+\mathrm{2}{q}+{p}=\mathrm{1}+{p}+{q}+{r}\Rightarrow\mathrm{2}{r}+{q}=\mathrm{1} \\…
Question Number 19982 by Tinkutara last updated on 19/Aug/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered}\:\mathrm{pairs} \\ $$$$\left({A},\:{B}\right)\:\mathrm{where}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{subsets}\:\mathrm{of} \\ $$$$\left\{\mathrm{1},\:\mathrm{2},\:…,\:\mathrm{5}\right\}\:\mathrm{such}\:\mathrm{that}\:\mathrm{neither}\:{A}\:\subseteq\:{B} \\ $$$$\mathrm{nor}\:{B}\:\subseteq\:{A}? \\ $$ Commented by mrW1 last updated on 22/Aug/17…
Question Number 19978 by Tinkutara last updated on 19/Aug/17 $$\mathrm{Let}\:{f}\:\mathrm{be}\:\mathrm{a}\:\mathrm{one}-\mathrm{to}-\mathrm{one}\:\mathrm{function}\:\mathrm{from} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{to}\:\mathrm{itself} \\ $$$$\mathrm{such}\:\mathrm{that}\:{f}\left({mn}\right)\:=\:{f}\left({m}\right){f}\left({n}\right)\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{natural}\:\mathrm{numbers}\:{m}\:\mathrm{and}\:{n}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{least}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{f}\left(\mathrm{999}\right)? \\ $$ Answered by mrW1 last updated…