Question Number 183845 by Michaelfaraday last updated on 30/Dec/22 Answered by MJS_new last updated on 31/Dec/22 $$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{{x}+\mathrm{2}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}}{\:\sqrt{\mathrm{2}}}\:\rightarrow\:{dx}=\frac{\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}}}{{t}}{dt}\right] \\ $$$$=\mathrm{2}\int\frac{{dt}}{\:\sqrt{\mathrm{2}}{t}^{\mathrm{2}} −\mathrm{2}{t}+\sqrt{\mathrm{2}}}=\mathrm{2arctan}\:\left(\sqrt{\mathrm{2}}{t}−\mathrm{1}\right)\:=…
Question Number 118307 by bramlexs22 last updated on 16/Oct/20 $$\:\:\:\underset{\mathrm{0}} {\int}^{\infty} \:\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$ Commented by Lordose last updated on 16/Oct/20 $$\mathrm{see}\:\mathrm{qst}\:\mathrm{118030}…
Question Number 118292 by Lordose last updated on 16/Oct/20 $$\boldsymbol{\mathrm{P}}\mathrm{rove}\:\mathrm{that}: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} −\mathrm{1}}{\mathrm{lnx}}\:=\:\boldsymbol{\mathrm{ln}}\mid\boldsymbol{\mathrm{n}}+\mathrm{1}\mid \\ $$ Answered by TANMAY PANACEA last updated on 16/Oct/20…
Question Number 118278 by Lordose last updated on 16/Oct/20 $$\mathrm{Evaluate} \\ $$$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{3}}} \mathrm{tan}^{\mathrm{2}} \mathrm{xsec}\left(\frac{\mathrm{x}}{\mathrm{3}}\right)\mathrm{dx} \\ $$$$\bigstar \\ $$ Answered by MJS_new last updated on…
Question Number 118270 by bramlexs22 last updated on 16/Oct/20 $$\:\:\:\int{x}\:{a}^{{x}} \:\left(\mathrm{1}−{a}\right)^{\mathrm{1}−{x}} \:{dx}? \\ $$ Commented by Dwaipayan Shikari last updated on 16/Oct/20 $$\left(\mathrm{1}−{a}\right)\int{x}\left(\frac{{a}}{\left(\mathrm{1}−{a}\right)}\right)^{{x}} \\ $$$$\frac{\mathrm{1}−{a}}{{log}\left(\frac{{a}}{\mathrm{1}−{a}}\right)}{x}\left(\frac{{a}}{\mathrm{1}−{a}}\right)^{{x}}…
Question Number 183806 by cortano1 last updated on 30/Dec/22 $$\:\:\int\:\frac{\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{25}}}}{{x}}\:{dx}\:=? \\ $$ Answered by Ar Brandon last updated on 30/Dec/22 $${I}=\int\frac{\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{25}}}}{{x}}{dx}\:,\:{x}=\mathrm{5sh}\vartheta \\ $$$$\:\:=\int\frac{\sqrt{\mathrm{5sh}\vartheta+\mathrm{5ch}\vartheta}}{\mathrm{5sh}\vartheta}\left(\mathrm{5ch}\vartheta{d}\vartheta\right)=\sqrt{\mathrm{5}}\int\left(\frac{{e}^{\mathrm{2}\vartheta}…
Question Number 118260 by bobhans last updated on 16/Oct/20 $$\int\:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$ Commented by bobhans last updated on 16/Oct/20 $${yes}…….\:{all}\:{sir} \\ $$…
Question Number 183794 by cortano1 last updated on 30/Dec/22 $$\:{If}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{cos}\:{x}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }\:{dx}=\:{T} \\ $$$$\:{then}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{{x}+\mathrm{1}}\:{dx}\:=\:?\: \\ $$ Commented by Frix last updated on…
Question Number 118246 by bemath last updated on 16/Oct/20 $$\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{ln}\:{x}}{{x}+\mathrm{1}}\:{dx}\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 16/Oct/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{logx}}{{x}+\mathrm{1}}{dx}=\left[{log}\left({x}\right){log}\left({x}+\mathrm{1}\right)\right]_{\mathrm{0}}…
Question Number 52703 by maxmathsup by imad last updated on 11/Jan/19 $${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}^{\mathrm{2}} \left({tx}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }\:{dx}\:\:{with}\:{t}\:\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\mathrm{2}{tx}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }\:{dx}…