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Category: Integration

x-x-x-x-x-dx-

Question Number 102894 by bramlex last updated on 11/Jul/20 $$\int\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+…}}}}}\:\mathrm{dx}\: \\ $$ Answered by bemath last updated on 11/Jul/20 $${y}=\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{…}}}}}\: \\ $$$${y}^{\mathrm{2}} \:=\:{x}+{y}\:\Rightarrow{y}^{\mathrm{2}} −{y}−{x}\:=\:\mathrm{0} \\…

let-a-gt-0-b-from-C-and-Re-b-gt-0-1-calculate-b-cos-ax-x-2-b-2-dx-2-find-the-value-of-x-sin-ax-x-2-b-2-dx-

Question Number 37357 by math khazana by abdo last updated on 12/Jun/18 $${let}\:{a}>\mathrm{0}\:{b}\:{from}\:{C}\:{and}\:\:{Re}\left({b}\right)>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{b}\:{cos}\left({ax}\right)}{{x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}\:{sin}\left({ax}\right)}{{x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} }\:{dx}.…

let-b-C-and-Re-b-gt-0-prove-that-e-iax-x-ib-dx-2ipi-e-ab-and-e-iax-x-ib-dx-0-

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} \\…