Category: Integration

Question Number 30773 by abdo imad last updated on 25/Feb/18 $${let}\:{a}>\mathrm{0}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}. \\…

Question Number 30767 by abdo imad last updated on 25/Feb/18 $${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−^{{n}} \sqrt{{x}}}{dx}. \\ $$ Commented by abdo imad last updated on 25/Feb/18…

Question Number 161839 by mathmax by abdo last updated on 23/Dec/21 $$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$ Answered by ArielVyny last updated on 23/Dec/21…

Question Number 30764 by abdo imad last updated on 25/Feb/18 $${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} {dt}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall{t}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$\frac{\mathrm{1}}{\:\sqrt{{e}}}\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \leqslant\:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} \:\leqslant\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{then}\:{find}\:{lim}_{{n}\rightarrow\infty\:} {I}_{{n}}…

Question Number 30761 by abdo imad last updated on 25/Feb/18 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:\:\:{with}\:{n}\:{from}\:{N}. \\ $$ Commented by abdo imad last updated on…