Question Number 221583 by MrGaster last updated on 08/Jun/25 Answered by MrGaster last updated on 08/Jun/25 $$=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{\mathrm{1}/\mathrm{11}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{\mathrm{9}/\mathrm{11}−\mathrm{1}} \beta\left(\frac{\:\mathrm{1}}{\mathrm{11}},\frac{\mathrm{9}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{9}}{\mathrm{11}}\right)\left(\int_{\mathrm{0}} ^{\mathrm{1}} {s}^{\mathrm{3}/\mathrm{11}−\mathrm{1}} \left(\mathrm{1}−{s}\right)^{\mathrm{5}/\mathrm{11}−\mathrm{1}} \beta\left(\frac{\mathrm{3}}{\mathrm{11}},\frac{\mathrm{5}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{11}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{11}}\right)\int_{\mathrm{0}}…
Question Number 221577 by Nicholas666 last updated on 08/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\mathrm{Prove};\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}\mathrm{d}{x}}{\left({x}^{\mathrm{2}} \:+\:\mathrm{1}\right)\left({e}^{\mathrm{2}\pi{x}} \:−\:\mathrm{1}\right)}\:=\:\frac{\gamma}{\mathrm{2}}\:−\:\frac{\mathrm{1}}{\mathrm{4}}\:\: \\ $$$$\:\:\:\:\mathrm{where};\:\gamma\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{Euler}'\mathrm{s}\:\mathrm{Mascheroni}\:\mathrm{constant}\:\:\:\: \\ $$$$ \\ $$ Answered by MrGaster…
Question Number 221601 by Nicholas666 last updated on 08/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\frac{{sin}\:\mathrm{2}{x}}{\mathrm{1}\:+\:{sin}\:\mathrm{3}{x}}\:{dx} \\ $$$$ \\ $$ Answered by Frix last updated on 08/Jun/25 $$=−\mathrm{2}\int\frac{\mathrm{cos}\:{x}\:\mathrm{sin}\:{x}}{\mathrm{4sin}^{\mathrm{3}} \:{x}\:−\mathrm{3sin}\:{x}\:+\mathrm{1}}{dx}\:\overset{\left[{t}=\mathrm{sin}\:{x}\right]}…
Question Number 221523 by zzjZZJ last updated on 07/Jun/25 Commented by MathematicalUser2357 last updated on 09/Jun/25 No result in range of all standard mathematical functions. I will define this integral as Q_221523(x). Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 221541 by PaulDirac last updated on 07/Jun/25 $$\int_{\mathrm{0}} ^{\:\pi} \mathrm{tan}\left(\mathrm{4}!\right)^{{e}^{\mathrm{3}{x}^{\mathrm{2}} +\:\mathrm{2}{x}} } .{dx} \\ $$ Answered by MrGaster last updated on 07/Jun/25 $$=\int_{\mathrm{0}}…
Question Number 221469 by Nicholas666 last updated on 06/Jun/25 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}; \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\left(\sqrt{{x}\:}\:\mathrm{ln}\:\sqrt{{x}}\right)^{\mathrm{2}} }{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} \:+\:\mathrm{4}{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}^{\mathrm{4}} \:+\:\mathrm{3}{x}^{\mathrm{6}} \:+\:\mathrm{2}{x}^{\mathrm{8}} \:+\:{x}^{\mathrm{10}} }\:\:\mathrm{d}{x} \\ $$$$\:\:\:\:\:\:\:=\:\frac{\mathrm{21}}{\mathrm{1024}}\:\zeta\left(\mathrm{3}\right)\:−\:\frac{\pi^{\mathrm{3}} }{\mathrm{1024}}\:+\:\frac{\pi^{\mathrm{3}}…
Question Number 221484 by Nicholas666 last updated on 06/Jun/25 $$ \\ $$$$\:\:\:{I}\:=\:\int_{\:\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{4}} } \:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} {y}^{\mathrm{2}} −{x}^{\mathrm{2}} {z}^{\mathrm{2}} −{x}^{\mathrm{2}} {w}^{\mathrm{2}} −{y}^{\mathrm{2}} {z}^{\mathrm{2}} −{y}^{\mathrm{2}} {w}^{\mathrm{2}} −{z}^{\mathrm{2}} {w}^{\mathrm{2}}…
Question Number 221454 by Nicholas666 last updated on 06/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove} \\ $$$$\:\:\underset{\:\mathrm{1}} {\int}^{\:+\infty} \:\frac{{x}}{\left(−\mathrm{1}\:+\:\mathrm{3}{x}^{\mathrm{2}} \:−\:\mathrm{3}{x}^{\mathrm{4}} \:+\:\mathrm{2}{x}^{\mathrm{6}} \right)\left(\mathrm{ln}\left({x}−\mathrm{1}\right)\:−\:\mathrm{2}\:\mathrm{ln}\:{x}\:+\:\mathrm{ln}\left({x}+\mathrm{1}\right)\right)}\:\:\mathrm{d}{x}\:=\:−\:\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{4}}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Answered by…
Question Number 221483 by Nicholas666 last updated on 06/Jun/25 $$ \\ $$$$\:\:\:\:{I}\left(\alpha\right)\:=\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \:\frac{\mathrm{ln}\left(\mathrm{1}\:+\:\alpha\left({xy}\:+\:{yz}\:+\:{zx}\right)\right)}{\mathrm{1}\:−\:{xyz}\:}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\alpha\:\in\:\left(\mathrm{0},\mathrm{1}\right) \\ $$$$ \\ $$ Commented by MrGaster last updated…
Question Number 221438 by Nicholas666 last updated on 05/Jun/25 $$ \\ $$$$\:\:\:\int_{\:\mathrm{1}} ^{\infty} \frac{\mathrm{ln}\left(\mathrm{ln}\:{x}\right)}{\mathrm{1}−\mathrm{2}{x}\:\mathrm{cos}\:\theta\:+\:{x}^{\mathrm{2}} }\:{dx}\:;\:\mathrm{for}\:\mathrm{all}\:\theta\:\in\:\left(−\pi\:,\:\pi\right)\:\:\:\:\: \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…