Question Number 215190 by MrGaster last updated on 31/Dec/24 $$\int\frac{{x}^{{p}−\mathrm{1}} }{\mathrm{1}+{x}^{{n}} }\mathrm{lnln}\frac{\mathrm{1}}{{x}}{dx},{p},{n}>\mathrm{0} \\ $$ Answered by MathematicalUser2357 last updated on 31/Dec/24 $$\mathrm{No}\:\mathrm{result}\:\mathrm{found}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{standard}\:\mathrm{mathematical}\:\mathrm{functions} \\ $$ Answered…
Question Number 215185 by MrGaster last updated on 31/Dec/24 $$\int_{−\infty} ^{\infty} \frac{\mathrm{exp}\left(\frac{{ax}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)\mathrm{exp}\left(\frac{{a}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx},{a}>\mathrm{0} \\ $$ Answered by MathematicalUser2357 last updated on 31/Dec/24 $$\mathrm{Oops}!\:\mathrm{Make}\:\mathrm{sure}\:\mathrm{you}\:\mathrm{typed}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{made}\:\mathrm{of}\:{x}.…
Question Number 215186 by MrGaster last updated on 31/Dec/24 $$\int_{\mathrm{0}} ^{\infty} {Ei}\left(−{x}\right)\mathrm{ln}{xdx} \\ $$ Answered by MathematicalUser2357 last updated on 31/Dec/24 $$=\mathrm{1}+\boldsymbol{\gamma}\:\left(\approx\mathrm{1}.\mathrm{57722}\right) \\ $$ Terms…
Question Number 215187 by MrGaster last updated on 31/Dec/24 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}\boldsymbol{{K}}^{\mathrm{2}} \left({x}\right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$ Answered by MathematicalUser2357 last updated on 31/Dec/24 $$\approx\mathrm{8}.\mathrm{473967} \\…
Question Number 215200 by mr W last updated on 31/Dec/24 $$\boldsymbol{\mathcal{HAPPY}}\:\:\boldsymbol{\mathcal{NEW}}\:\:\boldsymbol{\mathcal{YEAR}}\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\mathrm{9}} {\boldsymbol{\sum}}{n}}^{\mathrm{3}} \:! \\ $$ Commented by ajfour last updated on 01/Jan/25 $${Happy}\:{new}\:{Year}\:\bullet\bullet\mathrm{25}.\:{lets}\:{all}\:{be}\:\mathrm{25}{y} \\…
Question Number 215180 by hardmath last updated on 30/Dec/24 $$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{y}\:\:=\:\:\mathrm{31}}\\{\mathrm{y}^{\mathrm{2}} \:\:+\:\:\mathrm{x}\:\:=\:\:\mathrm{41}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\left(\mathrm{x}\:;\:\mathrm{y}\right)\:=\:? \\ $$ Commented by Ghisom last updated on 31/Dec/24 $$\mathrm{obviously} \\ $$$$\mathrm{5}^{\mathrm{2}} +\mathrm{6}=\mathrm{31}…
Question Number 215164 by MathematicalUser2357 last updated on 30/Dec/24 $$\mathrm{Let}\:\mathrm{the}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\:{x}^{\mathrm{2}} −\mathrm{12}{x}+{k}=\mathrm{0}\:\mathrm{is}\:\alpha,\:\alpha^{\mathrm{2}} , \\ $$$$\mathrm{Then}\:\mathrm{solve}\:\mathrm{for}\:\alpha\:\mathrm{and}\:{k}\:\mathrm{or}\:\mathrm{I}\:\mathrm{will}\:\mathrm{force}\:\mathrm{you}\:\mathrm{to}\:\mathrm{solve} \\ $$ Answered by A5T last updated on 30/Dec/24 $$\alpha+\alpha^{\mathrm{2}} =\mathrm{12}\Rightarrow\alpha=−\mathrm{4}\:\mathrm{or}\:\mathrm{3}…
Question Number 215166 by MathematicalUser2357 last updated on 30/Dec/24 $$\mathrm{If}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{8}{x}+\mathrm{2}{k}=\mathrm{0}\:\mathrm{has}\:\mathrm{two}\:\mathrm{different}\:\mathrm{negative}\:\mathrm{real}\:\mathrm{roots}, \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{k}\:\left(\mathrm{For}\:\mathrm{example},\:\mathrm{0}<{k}<\mathrm{3}\right)\:\mathrm{or}\:\mathrm{I}\:\mathrm{will}\:\mathrm{force}\:\mathrm{you}\:\mathrm{to}\:\mathrm{determine} \\ $$ Answered by A5T last updated on 30/Dec/24 $$\mathrm{x}_{\mathrm{1},\mathrm{2}} =\frac{−\mathrm{8}\underset{−} {+}\sqrt{\mathrm{64}−\mathrm{24k}}}{\mathrm{6}}=\frac{−\mathrm{8}\underset{−}…
Question Number 215167 by MathematicalUser2357 last updated on 30/Dec/24 $$\mathrm{If}\:\mathrm{the}\:\mathrm{quadratic}\:\mathrm{equation}\:{x}^{\mathrm{2}} −\mathrm{2}{x}−{m}+\mathrm{1}=\mathrm{0}\:\mathrm{has}\:\mathrm{two}\:\mathrm{different}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{roots}\:\alpha,\:\beta, \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{m}\:\left(\mathrm{For}\:\mathrm{example},\:\mathrm{0}<{m}<\mathrm{3}\right)\:\mathrm{or}\:\mathrm{I}\:\mathrm{will}\:\mathrm{force}\:\mathrm{you}\:\mathrm{to}\:\mathrm{determine} \\ $$ Answered by A5T last updated on 30/Dec/24 $$\mathrm{x}_{\mathrm{1},\mathrm{2}} =\frac{\mathrm{2}\underset{−} {+}\sqrt{\mathrm{4}−\mathrm{4}\left(\mathrm{1}−\mathrm{m}\right)}}{\mathrm{2}}=\mathrm{1}\underset{−}…
Question Number 215177 by Ismoiljon_008 last updated on 30/Dec/24 $$ \\ $$$$\:\:\:{x}^{\mathrm{3}} \:+\:{x}^{\mathrm{2}} \:+\:{x}\:=\:−\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\:\:\:\mathcal{F}{ind}\:{the}\:{value}\:{of}\:\:{x}\:\:!\: \\ $$$$ \\ $$$$\:\:\:\mathcal{H}{elp}\:{me},\:\:{please} \\ $$$$ \\ $$ Answered…