Question Number 128244 by mnjuly1970 last updated on 05/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:\:::\Omega=\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({sin}\left({x}\right)\right){d}=\frac{−\pi}{\mathrm{4}}{log}\left(\mathrm{2}\right)−\frac{{G}}{\mathrm{2}} \\ $$$$\:\:\:\:{log}\left(\mathrm{2}{sin}\left({x}\right)\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{−\mathrm{1}}{{n}}{cos}\left(\mathrm{2}{nx}\right) \\ $$$$\:\Omega=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left\{−{log}\left(\mathrm{2}\right)−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cos}\left(\mathrm{2}{nx}\right)}{{n}}\right\}{dx} \\…
Question Number 128221 by john_santu last updated on 05/Jan/21 $$\:{Given}\:{f}\left({x}+\frac{\mathrm{1}}{{x}}\right)\:=\:{x}^{\mathrm{4}} −\frac{\mathrm{1}}{{x}^{\mathrm{4}} }+\mathrm{2} \\ $$$$\:{then}\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{−\mathrm{2}} \right){f}\left({x}\right){dx}= \\ $$ Answered by liberty last updated on…
Question Number 128194 by Algoritm last updated on 05/Jan/21 Commented by MJS_new last updated on 05/Jan/21 $$\mathrm{it}'\mathrm{s}\:\mathrm{possible}\:\mathrm{by}\:\mathrm{using} \\ $$$$\mathrm{cos}\:{x}\:=\frac{\mathrm{e}^{\mathrm{i}{x}} +\mathrm{e}^{−\mathrm{i}{x}} }{\mathrm{2}}=\frac{\mathrm{e}^{\mathrm{2i}{x}} +\mathrm{1}}{\mathrm{2e}^{\mathrm{i}{x}} }\:\Rightarrow \\ $$$$\mathrm{8}\int\frac{{x}\mathrm{e}^{\mathrm{3i}{x}}…
Question Number 128192 by Algoritm last updated on 05/Jan/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 62653 by aliesam last updated on 23/Jun/19 $$\int\mathrm{x}\left(\mathrm{arctan}\left(\mathrm{x}\right)\right)^{\mathrm{2}} \:\mathrm{dx} \\ $$$$ \\ $$$$\int\frac{\mathrm{x}\:\mathrm{e}^{\mathrm{arctan}\left(\mathrm{x}\right)} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\mathrm{dx} \\ $$$$ \\ $$$$\int\frac{\mathrm{arcsin}\left(\mathrm{x}\right)}{\:\sqrt{\mathrm{1}+\mathrm{x}}}\:\mathrm{dx} \\ $$ Commented…
Question Number 62648 by Tawa1 last updated on 23/Jun/19 Commented by Tawa1 last updated on 23/Jun/19 $$\mathrm{Please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\:\mathrm{integrate}\:\mathrm{the}\:\mathrm{integral}.\:\mathrm{Thanks} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 128175 by bemath last updated on 05/Jan/21 $$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{dx}\:{dy}}{\mathrm{1}−{xy}^{\mathrm{3}} }\:? \\ $$ Commented by liberty last updated on 06/Jan/21 $$\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{xy}^{\mathrm{3}}…
Question Number 62613 by necx1 last updated on 23/Jun/19 Commented by Prithwish sen last updated on 23/Jun/19 $$=\int\frac{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\:\mathrm{dx}}{\mathrm{x}^{\mathrm{3}} \sqrt{\frac{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} }}}…
Question Number 62596 by aliesam last updated on 23/Jun/19 $$\int\mathrm{sin}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{cos}^{\mathrm{100}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$ Answered by MJS last updated on 23/Jun/19 $$\int\mathrm{sin}^{\mathrm{100}} \:{x}\:\mathrm{cos}^{\mathrm{100}} \:{x}\:{dx}=\int\left(\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\right)^{\mathrm{100}} {dx}=…
Question Number 128109 by bemath last updated on 04/Jan/21 $$\emptyset\:=\:\int\:\left(\mathrm{x}−\mathrm{2}\right)\:\sqrt{\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}}\:\mathrm{dx}\: \\ $$ Answered by liberty last updated on 04/Jan/21 $$\:\phi\:=\:\int\:\frac{\left(\mathrm{x}−\mathrm{2}\right)\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}{\mathrm{x}−\mathrm{1}}\:\mathrm{dx}\: \\ $$$$\:\mathrm{let}\:\mathrm{x}\:=\:\mathrm{sec}\:\mathrm{t}\: \\ $$$$\:\emptyset\:=\:\int\:\frac{\left(\mathrm{sec}\:\mathrm{t}−\mathrm{2}\right).\mathrm{tan}\:\mathrm{t}}{\mathrm{sec}\:\mathrm{t}−\mathrm{1}}.\:\left(\mathrm{sec}\:\mathrm{t}\:\mathrm{tan}\:\mathrm{t}\:\right)\:\mathrm{dt}\:…