Question Number 228329 by Danmola last updated on 07/Apr/26 $$\int\frac{{x}^{\mathrm{6}} {ln}^{\mathrm{2}} \left({x}\right)}{\left({e}^{\mathrm{2} } −\mathrm{1}\right)\left( \right.}\:{dx} \frac{\mathrm{5}}{\mathrm{96}}\:\frac{\zeta\left(\mathrm{3}\right)}{ }\:+\frac{\mathrm{1}}{ }\:\frac{\zeta\left(\mathrm{5}\right)}{ }\:−\frac{\mathrm{77}}{\mathrm{3840}}\:\frac{\zeta\left(\mathrm{7}\right)}{ }\:−\frac{\mathrm{11}}{\mathrm{1920}}\:\frac{\mathrm{1}}{ }\:−\frac{\mathrm{47}}{ }\:\frac{\mathrm{1}}{ }\: \\ $$$$…
Question Number 228113 by TonyCWX last updated on 25/Mar/26 $$\mathrm{Just}\:\mathrm{want}\:\mathrm{to}\:\mathrm{share}\:\mathrm{my}\:\mathrm{work}\:\mathrm{here}.\: \\ $$$$\mathrm{1}.\:\int\:{e}^{{x}} \mathrm{sin}\left({ax}\right)\:{dx} \\ $$$$\mathrm{Let}\:{I}\:=\:\int\:{e}^{{x}} \mathrm{sin}\left({ax}\right)\:{dx} \\ $$$$\overset{\mathrm{IBP}} {=}\:{e}^{{x}} \mathrm{sin}\left({ax}\right)\:−\:{a}\int\:{e}^{{x}} \mathrm{cos}\left({ax}\right)\:{dx} \\ $$$$\overset{\mathrm{IBP}} {=}\:{e}^{{x}} \mathrm{sin}\left({ax}\right)\:−\:{a}\left[{e}^{{x}}…
Question Number 228099 by TonyCWX last updated on 20/Mar/26 $${I}\:=\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\frac{\mathrm{2}{x}^{\mathrm{2}} }{\:\sqrt{\left(\mathrm{2}{x}−\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{2}\right)}}\:{dx} \\ $$$$=\:\mathrm{2}\int_{\mathrm{1}} ^{\mathrm{2}} \:\frac{{x}^{\mathrm{2}} }{\:\sqrt{\left(\mathrm{2}{x}−\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{2}\right)}}\:{dx} \\ $$$$\overset{{t}\:=\:\mathrm{2}{x}} {=}\:\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{2}} ^{\:\mathrm{4}} \:\frac{{t}^{\mathrm{2}} }{\:\sqrt{\left({t}−\mathrm{1}\right)\left({t}+\mathrm{2}\right)}}\:{dt} \\…
Question Number 228030 by TonyCWX last updated on 12/Mar/26 $${I}\:=\:\int\:\frac{\mathrm{1}}{{x}^{\mathrm{5}} +\mathrm{1}}\:{dx}\:=\:\int\:\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{4}} −{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −{x}+\mathrm{1}\right)}\:{dx} \\ $$$$=\:\int\:\left[\frac{\mathrm{1}}{\mathrm{5}\left({x}+\mathrm{1}\right)}\:−\:\frac{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{4}}{\mathrm{5}\left({x}^{\mathrm{4}} −{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −{x}+\mathrm{1}\right)}\right]\:{dx} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{5}}\int\:\frac{\mathrm{1}}{{x}+\mathrm{1}}\:{dx}\:−\:\frac{\mathrm{1}}{\mathrm{5}}\int\:\frac{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{4}}{{x}^{\mathrm{4}}…
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Question Number 227871 by Sid978 last updated on 22/Feb/26 Commented by Kassista last updated on 23/Feb/26 $${are}\:{there}\:{infinitely}\:{many}\:{integrals}?\:{only}\:{the}\:{last}\:{one} \\ $$$${has}\:{a}\:{variable}?\:{what}\:{do}\:{you}\:{mean}\:{with}\:{de}\:{dx}\:{as}\:{an}\:{exponent}? \\ $$ Terms of Service Privacy…
Question Number 227682 by Simurdiera last updated on 13/Feb/26 $$\mathrm{Help}\:\mathrm{please} \\ $$$${I}\:=\:\int_{\mathrm{0}} ^{\:\pi} \mathrm{cos}\left({x}\:+\:\mathrm{cos}\left({x}\right)\right)\:{dx} \\ $$ Commented by Tawa11 last updated on 13/Feb/26 https://youtu.be/m5eXP5oyDL4?si=iAaqjq-4F6BUkfoy Answered…
Question Number 227271 by Spillover last updated on 11/Jan/26 Answered by Kassista last updated on 11/Jan/26 $$ \\ $$$$\int\:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}−\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:=\:\int\:\frac{{x}^{\mathrm{2}} +\mathrm{5}}{{x}^{\mathrm{2}} +\mathrm{5}}\:{dx}\:−\int\frac{\mathrm{5}}{{x}^{\mathrm{2}}…
Question Number 227255 by Spillover last updated on 11/Jan/26 Answered by Spillover last updated on 11/Jan/26 Answered by Spillover last updated on 11/Jan/26 Answered by…
Question Number 227184 by Spillover last updated on 04/Jan/26 Answered by Spillover last updated on 04/Jan/26 Answered by Spillover last updated on 04/Jan/26 Answered by…