Category: Integration

Question Number 166916 by mnjuly1970 last updated on 02/Mar/22 $$$$$$\mathrm{\Omega }=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{H_n}{n(n+1)}=$$$$------$$$$\:\:\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\int_{\mathrm{0}\:} ^{\:\mathrm{1}} {x}^{\:{n}−\mathrm{1}} {ln}\left(\mathrm{1}−{x}\:\right){dx} \…

Question Number 101373 by Dwaipayan Shikari last updated on 02/Jul/20 $$\underset{n\to \mathrm{\infty }}{\lim }\underset{r=1}{\sum ^{4n}}\frac{\sqrt{n}}{\sqrt{r}{(3\sqrt{r}+4\sqrt{n})}^2}$$ Commented by Dwaipayan Shikari last updated on 02/Jul/20…

Question Number 35821 by prof Abdo imad last updated on 24/May/18 $$letf\left(t\right)={\int }_0^{\mathrm{\infty }}\frac{e^{-ax}-e^{-bx}}{x^2}{e}^{-{tx}^2}dxwitht>0$$$$1)calculate{f}^{{}^{\prime }}\left(t\right)$$$$\left.\mathrm{2}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right) \…

Question Number 101345 by bobhans last updated on 02/Jul/20 $$\left(1\right)\int \frac{\sec {}^{4}x\tan x}{\sec {}^{4}x+4}dx=$$$$\left(2\right)\int x^{2x}(2\ln x+2)dx=$$$$\left(3\right)\underset{0}{\stackrel{1}{\int }}\sqrt{1-x^2}dx=$$ Commented by john…

Question Number 166829 by mnjuly1970 last updated on 01/Mar/22 $$$$$$calculate$$$$\mathrm{I}f,f\left(x\right)=\frac{(x^2+1)\sqrt[3]{(x^2+x-2)(x^4-1)(x^2+2x-3)+16}+\sqrt{x^2+3}}{(1+x+x^2)}$$$$$$$$\:\:\:\:\:\:\:\:\:\:\:\:{then}\:,\:\:\:\:{f}\:'\:\left(\mathrm{1}\:\right)\:=?\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \…