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Category: Integration

advanced-calculus-prove-that-n-0-n-1-2-n-1-2-2-n-n-2pi-ln-2-

Question Number 133228 by mnjuly1970 last updated on 20/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{advanced}\:\:\:\:{calculus}…. \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\psi\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{2}^{{n}} .{n}!}=−\sqrt{\mathrm{2}\pi}\:\left(\gamma+{ln}\left(\mathrm{2}\right)\right)…. \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…

0-1-x-3-dx-x-1-3-3x-5-

Question Number 133222 by john_santu last updated on 20/Feb/21 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{3}} \:\mathrm{dx}}{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{3}} +\mathrm{3x}−\mathrm{5}} \\ $$ Answered by liberty last updated on 20/Feb/21 $$\:\mathrm{I}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{1}}…

let-f-a-0-dx-x-2-1-x-2-a-with-a-gt-0-1-determine-a-explicit-form-of-f-a-2-calculate-g-a-0-dx-x-2-1-x-2-a-2-3-give-f-n-a-at-form-of-integral-4-calcul

Question Number 67674 by Abdo msup. last updated on 30/Aug/19 $${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +{a}\right)}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+{a}\right)^{\mathrm{2}} } \\…