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…
Question Number 124825 by Study last updated on 06/Dec/20 Commented by Study last updated on 07/Dec/20 $${please}\:{help}\:{me}??? \\ $$ Commented by Study last updated on…
Question Number 59282 by Mr X pcx last updated on 07/May/19 $${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} {sin}\left(\left[{t}\right]\right){dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{3}\left[{t}\right]} {sin}\left(\left[{t}\right]\right){dt}\:. \\ $$ Commented…
Question Number 59279 by Mr X pcx last updated on 07/May/19 $${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{x}}{{sinx}}{dx} \\ $$ Commented by tanmay last updated on 07/May/19 $${f}\left({x}\right)=\frac{{x}}{{sinx}}\: \\…
Question Number 59277 by Mr X pcx last updated on 07/May/19 $${calculate}\:\int_{−\infty} ^{+\infty} \:\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx} \\ $$ Commented by maxmathsup by imad last updated…
Question Number 59278 by Mr X pcx last updated on 07/May/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }\:{dt} \\ $$$${determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\…