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Category: Integration

Advanced-Calculus-Let-f-R-Q-is-a-continuous-function-prove-that-f-is-a-constant-function-m-n-Adopted-from-mathematical-an

Question Number 165339 by mnjuly1970 last updated on 30/Jan/22 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:#\:{Advanced}\:\:\:{Calculus}\:#\:\:\: \\ $$$$\:\:\:\:\:\mathrm{L}{et}\:,\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{Q}\:\:{is}\:\:{a}\:\:{continuous} \\ $$$$\:\:\:\:\:\:{function}\:\:.\:{prove}\:{that}\:\:''\:{f}\:''\:{is}\:{a} \\ $$$$\:\:\:\:\:\:\:{constant}\:{function}\:.\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$\:\:\:\:\:\:\:\ast\:{Adopted}\:{from}\:{mathematical}\:{analysis}\:{book}\:\:\ast\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−−−−−−− \\ $$$$ \\…

let-give-a-gt-0-1-find-the-value-of-F-a-0-lnt-t-2-a-2-dt-2-find-the-value-of-G-a-0-aln-t-t-2-a-2-2-dt-3-find-the-value-of-0-ln-t-t-2-3-2-dt-

Question Number 34260 by math khazana by abdo last updated on 03/May/18 $${let}\:{give}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:{F}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{lnt}}{{t}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{G}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{aln}\left({t}\right)}{\left({t}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}}…

prove-n-N-3-n-1-n-3-n-1-3-n-2-3-

Question Number 165329 by mnjuly1970 last updated on 30/Jan/22 $$ \\ $$$$\:\:\:\:{prove}\:\:\:\:\:\:\:\:\:\:\left(\:{n}\in\:\mathbb{N}\:\right) \\ $$$$\:\:\:\:\mathrm{3}\left({n}+\mathrm{1}\right)\:\mid\:{n}^{\:\mathrm{3}} \:+\:\left({n}+\mathrm{1}\right)^{\:\mathrm{3}} +\:\left({n}+\mathrm{2}\:\right)^{\:\mathrm{3}} \\ $$$$ \\ $$ Answered by Rasheed.Sindhi last updated…