Question Number 223368 by Nicholas666 last updated on 22/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\boldsymbol{\mathrm{ln}}\left(\frac{\mathrm{2}\:\boldsymbol{\mathrm{cos}}\left({x}^{\mathrm{2}} \right)\:+\:\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left({x}/\mathrm{2}\right)}{\mathrm{1}\:+\:\boldsymbol{\mathrm{cos}}\:\left({x}/\mathrm{2}\right)}\right)\:\boldsymbol{\mathrm{d}}{x} \\ $$$$ \\ $$ Answered by MathematicalUser2357 last updated…
Question Number 223367 by Nicholas666 last updated on 22/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\mathrm{ln}\left(\mathrm{2}\:\mathrm{cos}\left({x}^{\mathrm{2}} \right)\:+\:\mathrm{ln}^{\mathrm{2}} \:\left(\frac{{x}}{\mathrm{2}}\right)\:\mathrm{d}{x}\right. \\ $$$$ \\ $$ Commented by MathematicalUser2357 last updated…
Question Number 223304 by Nadirhashim last updated on 21/Jul/25 $$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{{x}^{\mathrm{2}} } {\int}}\boldsymbol{{sin}}\left(\sqrt{\boldsymbol{{t}}}\right)\boldsymbol{{dt}}\:}{\boldsymbol{{x}}^{\mathrm{3}} }\:=…? \\ $$ Answered by Raphael254 last updated on 21/Jul/25 $$…
Question Number 223241 by Tawa11 last updated on 19/Jul/25 $$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}\:−\:\mathrm{x}\right)\:\mathrm{ln}\left(\mathrm{1}\:+\:\mathrm{x}\right)}{\mathrm{x}}\:\mathrm{dx} \\ $$ Answered by altarboy123 last updated on 19/Jul/25 $$=−\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{4}} \\ $$ Commented…
Question Number 223192 by Nicholas666 last updated on 17/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Evaluate}}\:;\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−{q}^{\mathrm{24}{n}} \right)\:\mathrm{d}{q} \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 223204 by Nadirhashim last updated on 17/Jul/25 $$\:\:\:\underset{\frac{\pi}{\mathrm{6}}} {\overset{\frac{\pi}{\mathrm{3}}} {\int}}\frac{\boldsymbol{{dx}}}{\mathrm{1}+\sqrt{\boldsymbol{{tanx}}}}\:=…? \\ $$ Answered by som(math1967) last updated on 17/Jul/25 $$\:\:\:{I}=\int_{\frac{\pi}{\mathrm{6}}\:} ^{\frac{\pi}{\mathrm{3}}} \frac{{dx}}{\mathrm{1}+\sqrt{{tan}\left(\frac{\pi}{\mathrm{3}}+\frac{\pi}{\mathrm{6}}−{x}\right)}} \\…
Question Number 223191 by Nicholas666 last updated on 17/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{Evaluate}}\:;\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\left(\mathrm{1}−{q}^{\mathrm{4}{n}} \right)^{\mathrm{6}} \:{dq} \\ $$$$ \\ $$ Terms of Service…
Question Number 223123 by Nicholas666 last updated on 15/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)\centerdot\mathrm{sec}\left({x}^{\mathrm{2}} \right)\:\mathrm{d}{x}\right. \\ $$$$ \\ $$ Terms of Service Privacy Policy…
Question Number 223116 by Nicholas666 last updated on 15/Jul/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{ln}^{\mathrm{5}} \left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\:{dx} \\ $$$$ \\ $$ Commented by MathematicalUser2357 last updated…
Question Number 223096 by fantastic last updated on 14/Jul/25 $$\underset{−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}} {\overset{\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}} {\int}}\:\:\frac{{x}^{\mathrm{4}} }{\mathrm{1}−{x}^{\mathrm{4}} }\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx}=? \\ $$ Answered by MathematicalUser2357 last updated on 22/Jul/25…