Question Number 124903 by Algoritm last updated on 06/Dec/20 Answered by MJS_new last updated on 06/Dec/20 $$\mathrm{just}\:\mathrm{decompose}\:\mathrm{and}\:\mathrm{solve},\:\mathrm{no}\:\mathrm{special} \\ $$$$\mathrm{knowledge}\:\mathrm{necessary}.\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}: \\ $$$$\frac{{x}−\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}+\frac{\mathrm{1}}{\mathrm{4}}\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)\:−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\mid{x}+\mathrm{1}\mid\:+{C} \\ $$$$\mathrm{now}\:\mathrm{try}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{this}\:\mathrm{for}\:\mathrm{yourself}…
Question Number 124888 by mnjuly1970 last updated on 06/Dec/20 $$:::::\:\:{prove}\:{that}\: \\ $$$$\:\:::::\:\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\infty} \frac{{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}=\frac{\pi}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$ Answered by mnjuly1970…
Question Number 124887 by mnjuly1970 last updated on 06/Dec/20 $$\:\:\:\:\:…{nice}\:\:{calculus}.. \\ $$$$\:\:\:{evaluate}\:: \\ $$$$\:\:\mathrm{2}\int_{\mathrm{1}} ^{\:\infty} \left(\frac{\left\{{x}\right\}−\frac{\mathrm{1}}{\mathrm{2}}}{{x}}\right){dx}−\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){dx}=??? \\ $$$$\left\{{x}\right\}:\:{fractional}\:{part}… \\ $$ Answered by Dwaipayan…
Question Number 190419 by horsebrand11 last updated on 02/Apr/23 $$\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\underset{−\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} {\overset{\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} {\int}}\:\mathrm{ln}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}\right){dx}\:{dy}\:=? \\ $$ Answered by witcher3 last updated…
Question Number 59344 by rahul 19 last updated on 08/May/19 $$\int\:{e}^{{x}} \left(\frac{\mathrm{1}−\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{cos}\:{x}}\right){dx}\:=\:? \\ $$ Answered by MJS last updated on 08/May/19 $$\mathrm{the}\:\mathrm{trick}\:\mathrm{is}\:\mathrm{this}: \\ $$$$\frac{\mathrm{1}−\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{cos}\:{x}}=\frac{\mathrm{1}}{\mathrm{1}−\mathrm{cos}\:{x}}−\frac{\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{cos}\:{x}}=\frac{\mathrm{1}}{\mathrm{2sin}^{\mathrm{2}} \:\frac{{x}}{\mathrm{2}}}−\frac{\mathrm{1}}{\mathrm{tan}\:\frac{{x}}{\mathrm{2}}}=…
Question Number 124853 by bemath last updated on 06/Dec/20 $$\:\:\int_{\mathrm{1}} ^{\:{x}^{\mathrm{3}} +\mathrm{5}{x}} {f}\left({t}\right)\:{dt}\:=\:\mathrm{2}{x}\:\: \\ $$$$\:{then}\:{f}\left(\mathrm{18}\right)\:=? \\ $$ Answered by liberty last updated on 06/Dec/20 $$\:\:\frac{{d}}{{dx}}\:\left[\:\int_{\:\mathrm{1}}…
Question Number 190385 by TUN last updated on 02/Apr/23 Answered by mehdee42 last updated on 02/Apr/23 $$\mathrm{2}\pi−{x}={u}\Rightarrow{dx}=−{du} \\ $$$${I}=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{ln}\left(−{sinu}+\sqrt{\mathrm{1}+{sin}^{\mathrm{2}} {u}}\right){du} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{2}\pi}…
Question Number 190371 by TUN last updated on 02/Apr/23 Answered by qaz last updated on 02/Apr/23 $$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{sin}\:\left({ln}\frac{\mathrm{1}}{{x}}\right)\frac{{x}^{{b}} −{x}^{{a}} }{{lnx}}{dx}=\int_{−\infty} ^{\mathrm{0}} \mathrm{sin}\:\left(−{u}\right)\centerdot\frac{{e}^{{ub}} −{e}^{{ua}} }{{u}}{e}^{{u}}…
Question Number 124827 by mnjuly1970 last updated on 06/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\:{nice}\:\:\:{calculus}\:… \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{log}\left(\mathrm{1}+{tan}\left({x}\right)\right)}{{tan}\left({x}\right)}{dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{48}}\:\checkmark \\ $$$$ \\ $$ Answered by mindispower last…
Question Number 190360 by mnjuly1970 last updated on 01/Apr/23 $$ \\ $$$${if}\:\:\:{x}+\:\frac{\mathrm{1}}{{x}}\:=\:\varphi\:\left(\:\:{Golden}\:{ratio}\right) \\ $$$$\:\:\:\:\:\Rightarrow\:\:\:{x}^{\:\mathrm{2000}} +\:\frac{\mathrm{1}}{{x}^{\:\mathrm{2000}} }=? \\ $$$$ \\ $$$$ \\ $$ Answered by Frix…