Question Number 67542 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({a}\:+{e}^{{ix}} \right)}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({a}\right)=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({a}+{e}^{{ix}} \right)^{\mathrm{2}} }…
Question Number 2004 by Yozzi last updated on 29/Oct/15 $${Suppose}\:\mathrm{0}<{b}\leqslant{a}.\:{Show}\:{that}\:{the}\:{area}\:{of} \\ $$$${intersection}\:{E}\cap{F}\:{of}\:{the}\:{two}\:{regions} \\ $$$${defined}\:{by}\: \\ $$$${E}=\left\{\left({x},{y}\right):\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\leqslant\mathrm{1}\right\}\:{and} \\ $$$${F}=\left\{\left({x},{y}\right):\frac{{x}^{\mathrm{2}} }{{b}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{a}^{\mathrm{2}}…
Question Number 67539 by mathmax by abdo last updated on 28/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{du}}{\mid{u}+{z}\mid^{\mathrm{2}} }\:\:{if}\:{z}\:={r}\:{e}^{{i}\theta} \:\:\:{and}\:−\pi<\theta<\pi \\ $$ Commented by ~ À ® @ 237…
Question Number 1997 by prakash jain last updated on 29/Oct/15 $${A}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…. \\ $$$${B}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…. \\ $$$${A}−{B}=? \\ $$ Commented by Rasheed Soomro last updated on 30/Oct/15…
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 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 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 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 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 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 ~…