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Category: Integration

sin-1-x-x-3-dx-

Question Number 137508 by EDWIN88 last updated on 03/Apr/21 $$\int\:\frac{\mathrm{sin}\:\left(\frac{\mathrm{1}}{{x}}\right)}{{x}^{\mathrm{3}} }\:{dx}\:=? \\ $$ Answered by liberty last updated on 03/Apr/21 $${L}=\int\:\frac{\mathrm{1}}{{x}}\mathrm{sin}\:\left(\frac{\mathrm{1}}{{x}}\right)\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)\:{dx} \\ $$$${let}\:{t}\:=\:\frac{\mathrm{1}}{{x}}\:\Rightarrow−{dt}\:=\:\frac{{dx}}{{x}^{\mathrm{2}} }…

advanced-calculus-prove-that-0-1-ln-x-ln-1-x-1-x-dx-13-8-3-pi-2-4-ln-2-

Question Number 137501 by mnjuly1970 last updated on 03/Apr/21 $$\:\:\:\:\:\:\:….{advanced}\:….\:{calculus}…. \\ $$$$\:\:{prove}\:{that}:: \\ $$$$\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left({x}\right).{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}+{x}}{dx}=\frac{\mathrm{13}}{\mathrm{8}}\zeta\left(\mathrm{3}\right)−\frac{\pi^{\mathrm{2}} }{\mathrm{4}}{ln}\left(\mathrm{2}\right)…. \\ $$ Answered by Ñï= last updated on…

nice-calculus-0-1-ln-ln-1-x-1-3-Im-

Question Number 137474 by mnjuly1970 last updated on 03/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:……..\:\:{calculus}….. \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\sqrt[{\mathrm{3}}]{{ln}\left(\sqrt{\mathrm{1}−{x}}\right)}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{Im}\left(\boldsymbol{\phi}\right)=??? \\ $$ Answered by mindispower last updated on 03/Apr/21…

Question-137461

Question Number 137461 by mathlove last updated on 03/Apr/21 Answered by Ñï= last updated on 03/Apr/21 $$\int\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{2}{x}+{x}^{\mathrm{3}} }{dx}=\frac{\mathrm{1}}{\mathrm{3}}\int\frac{\mathrm{2}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{x}+{x}^{\mathrm{3}} }{dx}=\frac{\mathrm{1}}{\mathrm{3}}{ln}\mid\mathrm{2}{x}+{x}^{\mathrm{3}} \mid+\frac{\mathrm{1}}{\mathrm{3}}\int\frac{{dx}}{{x}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}{ln}\mid\mathrm{2}{x}+{x}^{\mathrm{3}}…

0-3-4-dx-x-1-x-2-1-

Question Number 137448 by liberty last updated on 03/Apr/21 $$\int_{\mathrm{0}} ^{\:\mathrm{3}/\mathrm{4}} \frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\:? \\ $$ Answered by MJS_new last updated on 03/Apr/21 $$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}= \\…