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Category: Integration

evaluate-n-0-1-n-n-4-n-2-1-e-2-

Question Number 145200 by qaz last updated on 03/Jul/21 $$\mathrm{evaluate}::\:\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n}!\left(\mathrm{n}^{\mathrm{4}} +\mathrm{n}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{\mathrm{e}}{\mathrm{2}} \\ $$ Answered by mindispower last updated on 03/Jul/21 $${n}^{\mathrm{4}} +{n}^{\mathrm{2}}…

Question-79634

Question Number 79634 by TawaTawa last updated on 26/Jan/20 Commented by mathmax by abdo last updated on 27/Jan/20 $$\Omega\:=\int_{\frac{\pi}{\mathrm{5}}} ^{\frac{\mathrm{3}\pi}{\mathrm{10}}} \:\frac{{x}}{{sin}\left(\mathrm{2}{x}\right)}{dx}\:\:{changement}\:{x}=\frac{\pi}{\mathrm{2}}−{t}\:{givet}=\frac{\pi}{\mathrm{2}}−{x} \\ $$$$\Omega=\int_{\frac{\mathrm{3}\pi}{\mathrm{10}}} ^{\frac{\pi}{\mathrm{5}}} \:\frac{\frac{\pi}{\mathrm{2}}−{t}}{{sin}\left(\mathrm{2}{t}\right)}\left(−{dt}\right)\:=\frac{\pi}{\mathrm{2}}\int_{\frac{\pi}{\mathrm{5}}}…

1-expicite-f-x-0-1-ln-1-xt-2-1-t-2-dt-with-x-0-2-calculate-0-1-ln-1-t-2-1-t-2-dt-and-0-1-ln-1-2t-2-1-t-2-dt-

Question Number 79627 by mathmax by abdo last updated on 26/Jan/20 $$\left.\mathrm{1}\right)\:{expicite}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}}…

prove-that-with-using-hypergeometric-function-0-pi-sin-x-2-pi-3-3-1F-2-3-4-3-2-7-4-pi-4-4-

Question Number 79615 by M±th+et£s last updated on 26/Jan/20 $${prove}\:{that}\:{with}\:{using}\:{hypergeometric}\:{function} \\ $$$$\int_{\mathrm{0}} ^{\pi} {sin}\left({x}^{\mathrm{2}} \right)=\frac{\pi^{\mathrm{3}} }{\mathrm{3}}\:\mathrm{1}{F}_{\mathrm{2}} \left[\frac{\mathrm{3}}{\mathrm{4}};\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{7}}{\mathrm{4}};\frac{−\pi^{\mathrm{4}} }{\mathrm{4}}\right]\: \\ $$ Commented by mind is power…

dx-x-x-1-4-1-10-

Question Number 79612 by john santu last updated on 26/Jan/20 $$\int\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}\:}\left(\sqrt[{\mathrm{4}\:}]{\mathrm{x}}+\mathrm{1}\right)^{\mathrm{10}} }\:=\:? \\ $$ Answered by MJS last updated on 26/Jan/20 $$\int\frac{{dx}}{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}^{\frac{\mathrm{1}}{\mathrm{4}}} +\mathrm{1}\right)^{\mathrm{10}} }=…