Question Number 67530 by mathmax by abdo last updated on 28/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{n}−\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{2}{n}} }{dx}\:\:{with}\:{n}\geqslant\mathrm{3} \\ $$ Commented by ~ À ® @ 237…
Question Number 67531 by mathmax by abdo last updated on 28/Aug/19 $${prove}\:{that}\:{cos}\left(\pi{z}\right)\:=\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{{z}^{\mathrm{2}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}+{n}\right)^{\mathrm{2}} }\right) \\ $$ Commented by ~ À ® @ 237…
Question Number 67528 by mathmax by abdo last updated on 28/Aug/19 $${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{6}} }{dx} \\ $$ Commented by ~ À ® @ 237…
Question Number 67526 by mathmax by abdo last updated on 28/Aug/19 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dx}}{\mathrm{3}+\mathrm{2}{sinx}\:+{cosx}} \\ $$ Commented by mathmax by abdo last updated on 31/Aug/19…
Question Number 67527 by mathmax by abdo last updated on 28/Aug/19 $${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$$$ \\ $$ Commented by ~ À ®…
Question Number 67524 by mathmax by abdo last updated on 28/Aug/19 $${prove}\:{that}\:\forall{z}\:\in{C}\:\:{we}\:{have} \\ $$$${sinz}\:={z}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{{z}^{\mathrm{2}} }{{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\right) \\ $$ Commented by ~ À…
Question Number 67525 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{a}>{b}>\mathrm{0}\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{dx}}{\left({a}+{bsinx}\right)^{\mathrm{2}} } \\ $$ Commented by ~ À ® @ 237 ~…
Question Number 1988 by Rasheed Soomro last updated on 28/Oct/15 $${x}^{\mathrm{2}} =\:\frac{{f}\left({x}\right)+{f}\left(−{x}\right)}{\mathrm{2}} \\ $$$${f}\left({x}\right)=? \\ $$ Answered by 123456 last updated on 28/Oct/15 $$\mathrm{supossing}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{poly} \\…
Question Number 67522 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{z}\:{from}\:{C}−{Z}\:\:\:\:\:{prove}\:{that} \\ $$$$\frac{\pi}{{sin}\left(\pi{z}\right)}\:=\frac{\mathrm{1}}{{z}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \mathrm{2}{z}}{{z}^{\mathrm{2}} −{n}^{\mathrm{2}} }\:\:{and} \\ $$$$\frac{\pi{cos}\left(\pi{z}\right)}{{sin}\left(\pi{z}\right)}\:=\frac{\mathrm{1}}{{z}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{2}{z}}{{z}^{\mathrm{2}} −{n}^{\mathrm{2}} }…
Question Number 1987 by 123456 last updated on 28/Oct/15 $$\mathrm{lets}\:{a}<{b}\:\mathrm{and}\:{f}:\left[{a},{b}\right]\rightarrow\mathbb{R}\:\mathrm{integable}\:\mathrm{into}\:\left[{a},{b}\right]\:\mathrm{and}\:\mathrm{continuous} \\ $$$$\mathrm{lets}\:\mathrm{I}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{subset}\:\mathrm{of}\:\left[{a},{b}\right] \\ $$$$\mathrm{proof}\:\mathrm{that}\:\left(\mathrm{or}\:\mathrm{give}\:\mathrm{a}\:\mathrm{conter}\:\mathrm{example}\right) \\ $$$$\underset{\mathrm{I}} {\int}{fdx}=\mathrm{0}\:\forall\mathrm{I}\subset\left[{a},{b}\right]\Rightarrow{f}=\mathrm{0} \\ $$ Commented by prakash jain last updated…