Question Number 212052 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\int\frac{{dx}}{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} }= \\ $$$$\:\:\:\:\:\left[\mathrm{Ostrogradski}'\mathrm{s}\:\mathrm{M}\:\mathrm{ethod}\right] \\ $$$$=−\frac{{x}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{1}}= \\…
Question Number 212053 by universe last updated on 28/Sep/24 $$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\int_{\mathrm{0}} ^{\:\boldsymbol{{y}}} \:\boldsymbol{{e}}^{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} } \boldsymbol{{dx}}\right)\boldsymbol{{dy}}\:+\int_{\mathrm{1}} ^{\mathrm{2}} \left(\int_{\mathrm{0}} ^{\:\mathrm{2}−\boldsymbol{{y}}} \:\boldsymbol{{e}}^{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} }…
Question Number 212085 by hardmath last updated on 28/Sep/24 $$\mathrm{a},\mathrm{b},\mathrm{c}\:\in\:\mathbb{N} \\ $$$$\mathrm{5a}\:+\:\mathrm{6b}\:+\:\mathrm{7c}\:=\:\mathrm{70} \\ $$$$\mathrm{find}:\:\:\mathrm{max}\left(\mathrm{a}\right)\:=\:? \\ $$ Commented by Frix last updated on 28/Sep/24 $$\mathrm{If}\:\mathrm{0}\in\mathbb{N}\:\Rightarrow\:\mathrm{max}\:{a}\:=\mathrm{14}\:\:\:\:\:\left({b}={c}=\mathrm{0}\right) \\…
Question Number 212048 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{dx}}{\:\sqrt{{x}}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}}= \\ $$$$\:\:\:\:\:\left[{t}={x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right] \\…
Question Number 212049 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dx}}{\:\sqrt{−\mathrm{ln}\:{x}}}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{−\mathrm{ln}\:{x}}\right] \\ $$$$=−\mathrm{2}\underset{\infty} {\overset{\mathrm{0}} {\int}}\mathrm{e}^{−{t}^{\mathrm{2}}…
Question Number 212050 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\int\frac{{x}}{\:\sqrt{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{2}\sqrt{\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\right] \\ $$$$=\int{dt}={t}= \\…
Question Number 212083 by Spillover last updated on 28/Sep/24 Answered by Frix last updated on 28/Sep/24 $${I}=\int\:\frac{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}\:−\mathrm{1}}{\mathrm{2}}{dx}= \\ $$$$=−\frac{{x}+\mathrm{cos}\:{x}\:−\mathrm{sin}\:{x}}{\mathrm{2}}+{C} \\ $$ Commented by Spillover last…
Question Number 212051 by Spillover last updated on 28/Sep/24 Answered by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}}…
Question Number 212046 by Spillover last updated on 28/Sep/24 Answered by Spillover last updated on 28/Sep/24 Answered by Spillover last updated on 28/Sep/24 Answered by…
Question Number 212047 by Spillover last updated on 28/Sep/24 Answered by Spillover last updated on 28/Sep/24 Answered by Spillover last updated on 28/Sep/24 Terms of…