Question Number 158822 by qaz last updated on 09/Nov/21 $$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{arctan}\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{2n}+\mathrm{1}}=? \\ $$ Answered by mindispower last updated on 09/Nov/21 $$\underset{{n}\geqslant\mathrm{0}} {\sum}\left\{{arctan}\left(\frac{\mathrm{1}}{\mathrm{4}{n}+\mathrm{1}}\right)−{arctan}\left(\frac{\mathrm{1}}{\mathrm{4}{n}+\mathrm{3}}\right)\right\} \\…
Question Number 93265 by abdomathmax last updated on 12/May/20 $${calculate}\:\int_{\mathrm{0}} ^{\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\frac{{dx}}{\mathrm{3}+{sin}\left(\mathrm{3}{x}\right)} \\ $$ Commented by mathmax by abdo last updated on 12/May/20 $${A}\:=\int_{\mathrm{0}} ^{\frac{\mathrm{2}\pi}{\mathrm{3}}}…
Question Number 93248 by i jagooll last updated on 12/May/20 $$\int\underset{\mathrm{0}} {\overset{\mathrm{1}} {\:}}\:\mathrm{ln}\left(\mathrm{x}\right)\:\mathrm{dx}\: \\ $$ Commented by abdomathmax last updated on 12/May/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){dx}\:={lim}_{{a}\rightarrow\mathrm{0}^{+}…
Question Number 27693 by abdo imad last updated on 12/Jan/18 $$\left.\mathrm{1}\right)\:{calculate}\:\:\int\int_{\left.\right]\left.\mathrm{0}\left.,\left.\mathrm{1}\right]×\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]} \:\:\:\frac{{dxdy}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{t}}{{tant}}{dt}\:. \\ $$ Commented by abdo imad last updated…
Question Number 27692 by abdo imad last updated on 12/Jan/18 $${find}\:{by}\:{two}\:{ways}\:{the}\:{value}\:{of}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]} \:\:{x}^{{y}} \:\:{dxdxy}\:{then} \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{t}−\mathrm{1}}{{lnt}}{dt}\:\:. \\ $$ Commented by abdo imad last updated…
let-give-A-0-y-x-1-dxdxy-1-x-2-1-y-2-and-B-0-pi-4-ln-2cos-2-2cos-2-d-calculate-A-and-prove-that-B-A-
Question Number 27691 by abdo imad last updated on 12/Jan/18 $${let}\:{give}\:\:{A}=\int\int_{\mathrm{0}\leqslant{y}\leqslant{x}\leqslant\mathrm{1}} \:\:\:\:\:\:\frac{{dxdxy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\:\:{and} \\ $$$${B}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{ln}\left(\mathrm{2}{cos}^{\mathrm{2}} \theta\right)}{\mathrm{2}{cos}\left(\mathrm{2}\theta\right)}{d}\theta\:\:{calculate}\:{A}\:{and}\:{prove}\:{that}\:{B}={A}. \\ $$ Commented by abdo imad…
Question Number 93225 by Ajao yinka last updated on 11/May/20 Commented by prakash jain last updated on 12/May/20 $$\mathrm{1}+{x}+…+{x}^{{n}} \:\mathrm{has}\:\mathrm{no}\:\mathrm{root}\:\mathrm{in}\:\mathbb{R}\:\mathrm{for}\:{n}\:\mathrm{even} \\ $$$$\int_{−\infty} ^{\infty} {x}^{{n}} \delta\left(\mathrm{1}+{x}+…+{x}^{{n}}…
Question Number 27690 by abdo imad last updated on 12/Jan/18 $${find}\:\:\:{I}=\:\:\int\int_{{D}} {ln}\left(\mathrm{1}+{x}+{y}\right){dxdy}\:\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:\:\:/\:\:{x}+{y}\leqslant\mathrm{1}\:{and}\:{x}\geqslant\mathrm{0}\:{and}\:{y}\geqslant\mathrm{0}\:\right\}. \\ $$ Commented by abdo imad last updated on 14/Jan/18…
Question Number 93227 by Ajao yinka last updated on 11/May/20 Commented by prakash jain last updated on 12/May/20 $$\mathrm{2}\int_{−\infty} ^{+\infty} {x}^{\mathrm{4}} \delta\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right){dx}=\mathrm{2}×\left(\frac{\mathrm{2}}{\mathrm{4}}\right)=\mathrm{1} \\…
Question Number 27684 by abdo imad last updated on 12/Jan/18 $$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{the}\:{integral} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{cosx}\right)}{{cosx}}{dx} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{I}=\:\int\int_{{D}} \:\:\frac{{siny}}{\mathrm{1}+{cosx}\:{cosy}}{dxdy}\:{with}\: \\ $$$${D}=\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right]^{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{I}. \\ $$ Commented…