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Category: Differentiation

1-prove-that-0-e-t-ln-t-1-e-t-2-dt-1-2-ln-pi-2-

Question Number 151819 by mnjuly1970 last updated on 23/Aug/21 $$ \\ $$$$\:\:\:\:\:\mathrm{1}.\:{prove}\:{that}\:: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{\:{e}^{\:{t}} .{ln}\left({t}\:\right)}{\left(\mathrm{1}\:+\:{e}^{\:{t}} \right)^{\:\mathrm{2}} }\:{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left({ln}\left(\frac{\pi}{\mathrm{2}}\:\right)−\:\gamma\:\right)\: \\ $$$$\:\:\: \\ $$ Answered by…

Find-y-CF-PI-in-following-differential-equation-d-2-y-dx-2-3-dy-dx-2y-e-2x-sinx-

Question Number 86242 by niroj last updated on 27/Mar/20 $$\:\boldsymbol{\mathrm{Find}}\:\:\boldsymbol{\mathrm{y}}=\boldsymbol{\mathrm{C}}\mathrm{F}+\boldsymbol{\mathrm{P}}\mathrm{I}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{following}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}}: \\ $$$$\:\:\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}} }+\mathrm{3}\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}+\mathrm{2}\boldsymbol{\mathrm{y}}=\:\boldsymbol{\mathrm{e}}^{\mathrm{2}\boldsymbol{\mathrm{x}}} \:\boldsymbol{\mathrm{sinx}}\:. \\ $$$$\: \\ $$$$ \\ $$ Answered by TANMAY PANACEA.…

show-that-F-0-sin-4-x-2-x-2-dx-1-8-4-2-pi-m-n-

Question Number 151744 by mnjuly1970 last updated on 22/Aug/21 $$ \\ $$$$\:\:\:\:\:\:\:\:{show}\:\:{that}…. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathscr{F}\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{4}} \:\left({x}^{\:\mathrm{2}} \:\right)\:}{{x}^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\mathrm{1}}{\mathrm{8}}\:\left(\:\mathrm{4}\:−\:\sqrt{\mathrm{2}}\:\right)\sqrt{\pi}\:…..\blacksquare\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:…{m}.{n}… \\ $$$$…

1-5-4x-2x-2-dx-

Question Number 86092 by ar247 last updated on 27/Mar/20 $$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} }}{dx} \\ $$ Commented by abdomathmax last updated on 27/Mar/20 $${I}\:=\int\:\:\frac{{dx}}{\:\sqrt{−\mathrm{2}{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{5}}}\:\:{we}\:{have} \\ $$$${I}=\int\:\:\:\frac{{dx}}{\:\sqrt{\mathrm{5}−\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}−\mathrm{1}\right)}}=\int\:\:\frac{{dx}}{\:\sqrt{\mathrm{5}−\mathrm{2}\left({x}+\mathrm{1}\right)^{\mathrm{2}}…