Question Number 59576 by maxmathsup by imad last updated on 12/May/19 $${let}\:{f}\left({x}\right)\:=\int\:\:\:\:\:\:\:\frac{{dt}}{\left({x}+{t}\right)\sqrt{{t}^{\mathrm{2}} −{x}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:\int\:\:\:\:\:\frac{{dt}}{\left({x}+\mathrm{2}\right)\sqrt{{t}^{\mathrm{2}} −\mathrm{4}}}\:\:{and}\:\:\int\:\:\:\:\:\:\frac{{dt}}{\left({x}+\mathrm{1}\right)\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}} \\ $$ Commented by tanmay…
Question Number 59575 by maxmathsup by imad last updated on 12/May/19 $${find}\:\int\:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} {x}}{dx} \\ $$ Answered by Smail last updated on 12/May/19 $$\int\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} {x}}{dx}=\int\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\frac{\mathrm{1}+{cos}\mathrm{2}{x}}{\mathrm{2}}}{dx} \\…
Question Number 59563 by aliesam last updated on 11/May/19 Commented by aliesam last updated on 11/May/19 $${thank}\:{you}\:{sir} \\ $$ Answered by MJS last updated on…
Question Number 125098 by mnjuly1970 last updated on 08/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}\:… \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:::\:{Apery}'{s}\:{constant} \\ $$$$\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} {x}^{\mathrm{2}^{\mathrm{2}} } +\mathrm{4}^{\mathrm{3}} {x}^{\mathrm{2}^{\mathrm{3}} } +…\right)\frac{{ln}^{\mathrm{2}} \left({x}\right)}{{x}\left(\mathrm{1}+{x}\right)}\right\}{dx} \\…
Question Number 125096 by mnjuly1970 last updated on 08/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:\:{calculus}… \\ $$$$\:\:\:\:{suppose}\:::\:{z}\:={x}−{iy}\:\:\&\:\sqrt[{\mathrm{3}}]{{z}}\:={p}+{iq} \\ $$$$\:\:\:{then}\:\:{find}\:::\:\:\:{A}=\frac{\frac{{x}}{{p}}+\frac{{y}}{{q}}}{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} }\:=?? \\ $$$$\:{note}\::\:{i}=\sqrt{−\mathrm{1}} \\ $$ Answered by MJS_new last updated…
Question Number 190624 by Rupesh123 last updated on 07/Apr/23 Answered by mr W last updated on 07/Apr/23 Commented by mr W last updated on 07/Apr/23…
Question Number 190623 by mnjuly1970 last updated on 07/Apr/23 $$ \\ $$$$\:\:\:\:\:\:\mathrm{calculate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\right)^{\:{n}−\mathrm{1}} }{{n}}\:\mathrm{cos}\:\left(\frac{\:{n}\pi}{\mathrm{3}}\:\right)\:=? \\ $$$$ \\ $$ Answered by…
Question Number 190622 by mnjuly1970 last updated on 07/Apr/23 $$ \\ $$$$\:\:\:{prove}\:: \\ $$$$\:\:\int_{−\infty} ^{\:\infty} \:\:\:\left(\frac{\:{x}}{\left.\:\underline{\vdots} \right)^{\mathrm{2}} \mathrm{d}{x}=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{k}\:^{\mathrm{2}} }\:\:\:\lessdot}\right. \\ $$ Answered by…
Question Number 59528 by Mr X pcx last updated on 11/May/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\mathrm{1}+{xch}\left({t}\right)}\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left(\mathrm{1}+{xch}\left({t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{1}+\mathrm{3}{ch}\left({t}\right)}\:{and}\:…
Question Number 59526 by Mr X pcx last updated on 11/May/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{2}{sh}\left({x}\right)+\mathrm{3}{ch}\left({x}\right)} \\ $$ Commented by maxmathsup by imad last updated on 12/May/19…