Question Number 55930 by ajfour last updated on 06/Mar/19 $$\int_{\mathrm{0}} ^{\:\:\pi} \frac{{x}\mathrm{tan}\:{x}}{\mathrm{sec}\:{x}+\mathrm{tan}\:{x}}{dx}\:=\:\left({is}\:{it}\:\frac{\pi^{\mathrm{2}} }{\mathrm{2}}−\pi\right)? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 06/Mar/19 $$\int_{\mathrm{0}} ^{\pi} \frac{{xsinx}}{\mathrm{1}+{sinx}}{dx}…
Question Number 186973 by manlikeAkin last updated on 12/Feb/23 $$\int\mathrm{5}{x} \\ $$ Commented by mr W last updated on 12/Feb/23 $${there}\:{exists}\:{no}\:{such}\:{a}\:{thing}\:\int\mathrm{5}{x}\:{in} \\ $$$${mathematics}! \\ $$…
Question Number 55873 by Easyman32 last updated on 05/Mar/19 $${Integrate}..\int\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{{x}}}}\:{dx} \\ $$ Commented by maxmathsup by imad last updated on 05/Mar/19 $${let}\:{I}\:=\int\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{{x}}}}\:{dx}\:\:\:{changement}\:\sqrt{\mathrm{1}+\sqrt{{x}}}={t}\:{give}\:\mathrm{1}+\sqrt{{x}}={t}^{\mathrm{2}} \:\Rightarrow \\ $$$$\sqrt{{x}}={t}^{\mathrm{2}}…
Question Number 121397 by abdelsalamalmukasabe last updated on 07/Nov/20 Answered by MJS_new last updated on 07/Nov/20 $$\int\mathrm{2}^{\mathrm{1}/{x}} {dx}= \\ $$$$\:\:\:\:\:\left[\mathrm{by}\:\mathrm{parts}\right] \\ $$$$=\mathrm{2}^{\mathrm{1}/{x}} {x}+\mathrm{ln}\:\mathrm{2}\:\int\frac{\mathrm{2}^{\mathrm{1}/{x}} }{{x}}{dx}= \\…
Question Number 55855 by Joel578 last updated on 05/Mar/19 $$\mathrm{How}\:\mathrm{to}\:\mathrm{integrate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{sec}^{\mathrm{2}} \:{x}}{{x}\sqrt{{x}}}\:{dx}\:\:? \\ $$ Commented by tanmay.chaudhury50@gmail.com last updated on 05/Mar/19 Terms of Service…
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Question Number 121384 by john santu last updated on 07/Nov/20 $$\:\int\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\:\mathrm{dx}\: \\ $$ Commented by MJS_new last updated on 07/Nov/20 $$\mathrm{not}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{using}\:\mathrm{only}\:\mathrm{elementary} \\ $$$$\mathrm{calculus}.\:\mathrm{I}\:\mathrm{guess}\:\mathrm{we}\:\mathrm{need}\:\mathrm{to}\:\mathrm{substitute}\:\mathrm{and} \\…
Question Number 186911 by Spillover last updated on 11/Feb/23 $$ \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\mathrm{1}+{a}^{{x}} +{a}^{\frac{{x}}{\mathrm{2}}} }{dx}\:=\:\frac{\mathrm{1}}{\mathrm{ln}\:{a}}\left[\mathrm{ln}\:\mathrm{3}−\frac{\pi}{\mathrm{3}\sqrt{\mathrm{3}}}\right] \\ $$ Answered by witcher3 last updated on 13/Feb/23…
Question Number 186910 by Spillover last updated on 11/Feb/23 $$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{{a}} \sqrt{\frac{\mathrm{cos}\:\mathrm{2}{x}−\mathrm{cos}\:\mathrm{2}{a}}{\mathrm{cos}\:\mathrm{2}{x}+\mathrm{1}}}\:{dx}=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\mathrm{cos}\:{a}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$ Answered by witcher3…
Question Number 55834 by Tawa1 last updated on 04/Mar/19 Answered by ajfour last updated on 04/Mar/19 $$\:\:\:\left(\mathrm{2}−\mathrm{1}\right){f}_{{min}} \left({x}=\mathrm{2}\right)\leqslant\int_{\mathrm{1}} ^{\:\:\mathrm{2}} {f}\left({x}\right){dx}\:\leqslant\:\frac{\left(\mathrm{2}−\mathrm{1}\right)}{\mathrm{2}}\left[{f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)\right] \\ $$$$\:\:\:\Rightarrow\:\frac{\mathrm{1}}{\mathrm{17}}\:\leqslant\int_{\mathrm{1}} ^{\:\:\mathrm{2}} \frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} }\:\leqslant\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{17}}\right)\:<\:\frac{\mathrm{7}}{\mathrm{24}}\:.…