Category: Integration

Question Number 48720 by Abdo msup. last updated on 27/Nov/18 $$calculate{\int }_0^{\mathrm{\infty }}\frac{x^2-2cosx+1}{x^4+x^2+1}dx$$ Commented by Abdo msup. last updated on…

Question Number 48718 by Abdo msup. last updated on 27/Nov/18 $$let{I}_n={\int }_0^1{(1-t^2)}^{n}dt$$$$1)calculate{I}_{n}byrecurrence$$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{n}} ^{{k}}…

Question Number 48703 by cesar.marval.larez@gmail.com last updated on 27/Nov/18 Answered by tanmay.chaudhury50@gmail.com last updated on 27/Nov/18 $$3)\int {sin}^3\left(2x\right)cos\left(2x\right)dx$$$$t=sin2xdt=2cos2xdx$$$$=\frac{1}{2}\int t^{3}dt$$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{t}^{\mathrm{4}}…

Question Number 48643 by cesar.marval.larez@gmail.com last updated on 26/Nov/18 Commented by maxmathsup by imad last updated on 26/Nov/18 $$1)letI=\int \frac{3x+2}{(x-1)(x-2)(x^2+3)}dxletdecomposeF\left(x\right)=\frac{3x+2}{(x-1)(x-2)(x^2+3)}$$$${F}\left({x}\right)\:=\frac{{a}}{{x}−\mathrm{1}}\:+\frac{{b}}{{x}−\mathrm{2}}\:+\frac{{cx}+{d}}{{x}^{\mathrm{2}\:} +\mathrm{3}} \…

Question Number 114146 by Eric002 last updated on 17/Sep/20 $$prove$$$${\int }_0^1\frac{t^{n+2}\varphi (t,1,n+2)+ln(1-t)+t{H}_{n+1}}{t(t-1)}dt$$$$=\frac{H_{n+1}^{\left(2\right)}-{\left(H_n\right)}^2}{2}$$ Commented…