Question Number 126133 by I want to learn more last updated on 17/Dec/20 Answered by Dwaipayan Shikari last updated on 17/Dec/20 $$\int_{−\mathrm{2}} ^{\mathrm{2}} \left({x}^{\mathrm{3}} {cos}\frac{{x}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}}…
Question Number 60595 by maxmathsup by imad last updated on 22/May/19 $${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−{ax}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−\left({cos}\theta\right){x}\right)^{\mathrm{2}} }{dx}\:\:{with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\…
Question Number 60586 by Mr X pcx last updated on 22/May/19 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}^{\mathrm{2}} \left({x}\right)}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$ Commented by maxmathsup by imad last…
Question Number 126073 by benjo_mathlover last updated on 17/Dec/20 Commented by benjo_mathlover last updated on 17/Dec/20 $${The}\:{graph}\:{of}\:{the}\:{differentiable}\: \\ $$$${function}\:{g}\:{with}\:{domain}\:−\mathrm{6}\leqslant{x}\leqslant\mathrm{2}\:{is} \\ $$$${shown}\:{in}\:{the}\:{figure}\:{above}.\:{The}\:{areas} \\ $$$${of}\:{the}\:{regions}\:{bounded}\:{by}\:{the}\:{x}−{axis} \\ $$$${and}\:{the}\:{graph}\:{of}\:{g}\:{on}\:{the}\:{intervals}\:\left[−\mathrm{6},−\mathrm{5}\right]…
Question Number 60534 by aliesam last updated on 21/May/19 Commented by maxmathsup by imad last updated on 22/May/19 $${we}\:{have}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{e}^{{x}^{\mathrm{2}} } }\:=\mathrm{2}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}}…
Question Number 126068 by benjo_mathlover last updated on 17/Dec/20 Answered by liberty last updated on 17/Dec/20 $${h}\left({x}\right)=\int_{\mathrm{1}} ^{\:{x}} {f}\left({t}\right){dt}\:\Leftrightarrow\:{h}\left(\mathrm{1}\right)=\int_{\mathrm{1}} ^{\:\mathrm{1}} {f}\left({t}\right){dt}\:=\:\mathrm{0} \\ $$ Terms of…
Question Number 126065 by Lordose last updated on 17/Dec/20 $$ \\ $$$$\mathrm{Show}\:\mathrm{that}:: \\ $$$$\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}}\mathrm{dx}\:=\:−\frac{\mathrm{3}}{\mathrm{16}}\zeta\left(\mathrm{4}\right) \\ $$$$\mathrm{Goodluck} \\ $$ Answered by mnjuly1970 last…
Question Number 60506 by prof Abdo imad last updated on 21/May/19 $${calculate}\:\int\int_{{W}} \:\:\:\:\:\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }}{{x}+{y}}\:{dxdy} \\ $$$${with}\:{W}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}<{x}<\mathrm{1}\:{and}\:\mathrm{0}<{y}<\mathrm{1}.\right. \\ $$ Commented by Mr X pcx…
Question Number 60496 by maxmathsup by imad last updated on 21/May/19 $${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{lnx}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }{dx} \\ $$ Commented by Mr X pcx last updated on…
Question Number 60498 by abdo mathsup 649 cc last updated on 21/May/19 $${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{{t}\:+{x}\:+{x}^{\mathrm{2}} }{dx}\:\:{with}\:{t}\:\geqslant\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{3}} \:\:\:\frac{{dx}}{\:\sqrt{{t}+{x}\:+{x}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}}…