Category: Integration

Question Number 123703 by Algoritm last updated on 27/Nov/20 Answered by MJS_new last updated on 27/Nov/20 $$=\int {\left(\cos x\right)}^{5/3}dx$$$$\mathrm{use}\mathrm{these}\mathrm{formulas}:$$$$\int\left(\mathrm{sin}\:{x}\right)^{{p}/{q}} {dx}=\frac{{q}}{{p}+{q}}\left(\mathrm{sin}\:{x}\right)^{\frac{{p}+{q}}{{q}}} \:_{\mathrm{2}} \mathrm{F}_{\mathrm{1}}…

Question Number 123687 by mnjuly1970 last updated on 27/Nov/20 $$\dots nicecalculus..$$$$provethat::{lim}_{x\to 0}(\frac{2\varphi \left(x\right)}{x^2}+\frac{{\pi }^2}{3x})\stackrel{???}{=}\zeta \left(3\right)$$$$where$$$$\varphi \left(x\right)={\int }_0^1\frac{(t^x-1)\left(ln(1-t)\right)}{tln\left(t\right)}dt$$…

Question Number 123674 by mathmax by abdo last updated on 27/Nov/20 $$\mathrm{find}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-2\mathrm{x}}\ln (1+{\mathrm{e}}^{\mathrm{x}})\mathrm{dx}$$ Answered by Dwaipayan Shikari last updated on…

Question Number 189189 by MikeH last updated on 13/Mar/23 Answered by cortano12 last updated on 13/Mar/23 $$\mathrm{it}\mathrm{should}\mathrm{be}\underset{-1/2}{\stackrel{1/2}{\int }}\sqrt{{\mathrm{x}}^2+1+\sqrt{{\mathrm{x}}^4+{\mathrm{x}}^2+1}}\mathrm{dx}$$$$=\:\underset{−\mathrm{1}/\mathrm{2}} {\overset{\mathrm{1}/\mathrm{2}}…

Question Number 189144 by talminator2856792 last updated on 12/Mar/23 $$$$$$$$$${\int }_0^1{\int }_0^1{\int }_0^1\frac{\sqrt{x+y+z}}{\sqrt{x}+\sqrt{y}+\sqrt{z}}dxdydz$$$$$$$$\: \…