Category: Integration

Question Number 33346 by prof Abdo imad last updated on 14/Apr/18 $$letaandbfromR/a<bf[a,b]\to Ccontinje$$$$provethat\mathrm{\forall }n\in N{\int }_a^b\left(\prod _{k=0}^{n-1}(x+k)\right)f\left(x\right)dx=0$$$$\Rightarrow f=0$$ Terms of Service…

Question Number 33345 by prof Abdo imad last updated on 14/Apr/18 $$forx\in ]0,+\mathrm{\infty }[let\psi \left(x\right)=\frac{{\mathrm{\Gamma }}^{{}^{\prime }}\left(x\right)}{\mathrm{\Gamma }\left(x\right)}$$$$1)provethat\psi \left(x\right)=-\frac{1}{x}-\gamma +x\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n(x+n)}$$$$2)ptovethat\gamma =-{\mathrm{\Gamma }}^{{}^{\prime }}\left(1\right)$$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {ln}\left({x}\right){dx}\:=−\gamma\:.…

Question Number 33344 by prof Abdo imad last updated on 14/Apr/18 $$provethat\mathrm{\forall }\alpha \in ]0,+\mathrm{\infty }[$$$${lim}_{n\to \mathrm{\infty }}{\int }_0^n{(1-\frac{x}{n})}^n{x}^{\alpha -1}dx=\mathrm{\Gamma }\left(\alpha \right).$$ Commented by prof Abdo…

Question Number 33342 by prof Abdo imad last updated on 14/Apr/18 $$let{I}_n={\int }_0^1\frac{1}{x}(1-{(1-\frac{x}{n})}^n)dxand$$$$J_n={\int }_1^n\frac{1}{x}{(1-\frac{x}{n})}^{n}dx,nintegrnot0$$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{lim}_{} \:{I}_{{n}}…

Question Number 33340 by prof Abdo imad last updated on 14/Apr/18 $$findaequivalentof$$$$A_n={\int }_0^n\sqrt{1+{(1-\frac{x}{n})}^n}dx(n\to \mathrm{\infty })$$ Commented by prof Abdo imad…