Category: Integration

Question Number 37356 by math khazana by abdo last updated on 12/Jun/18 $${let}\:\:{b}\:\in{C}\:\:{and}\:{Re}\left({b}\right)\:>\mathrm{0}\:{prove}\:{that} \\ $$$$\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{iax}} }{{x}−{ib}}{dx}\:=\mathrm{2}{i}\pi\:{e}^{−{ab}\:\:} \:\:\:{and} \\ $$$$\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{iax}} }{{x}+{ib}}\:{dx}\:=\mathrm{0} \\…

Question Number 37352 by math khazana by abdo last updated on 12/Jun/18 $${let}\:\:{f}\left({z}\right)\:=\:{e}^{−\frac{\mathrm{1}}{{z}^{\mathrm{2}} }} \:\: \\ $$$$\left.\mathrm{1}\right)\:{give}\:{f}\left({z}\right)\:{at}\:{form}\:{of}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{give}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {f}\left({z}\right){dz}\:\:\:{at}\:{form}\:{of}\:{serie}\:. \\ $$ Commented by…

Question Number 37349 by math khazana by abdo last updated on 12/Jun/18 $${calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{\mathrm{1}−\mathrm{2}{pcost}\:+{p}^{\mathrm{2}} }\:\:{if}\:\mid{p}\mid<\mathrm{1} \\ $$ Commented by math khazana by abdo last…

Question Number 37347 by math khazana by abdo last updated on 12/Jun/18 $${let}\:{r}\:=\sqrt{{p}^{\mathrm{2}} \:+{q}^{\mathrm{2}} }\:\:\:{p}\:{and}\:{q}\:{from}\:{R}\:\:{and}\:{p}>\mathrm{0}\:\:{q}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:{e}^{−{px}} \:\frac{{cos}\left({px}\right)}{\:\sqrt{{x}}}{dx}=\frac{\sqrt{\pi}}{{r}}\sqrt{\frac{{r}+{p}}{\mathrm{2}}} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{px}} \:\:\frac{{sin}\left({qx}\right)}{\:\sqrt{{x}}}{dx}\:=\frac{\sqrt{\pi}}{{r}}\:\sqrt{\frac{{r}−{p}}{\mathrm{2}}}…

Question Number 37346 by math khazana by abdo last updated on 12/Jun/18 $${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{a}\:{cos}^{\mathrm{2}} {t}\:+{b}\:{sin}^{\mathrm{2}} {t}} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:. \\ $$ Terms of Service Privacy…