Menu Close

Category: Integration

advanced-calculus-prove-that-0-x-2-cosh-2-x-2-dx-2-2-4-pi-1-2-

Question Number 140715 by mnjuly1970 last updated on 11/May/21 $$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:……{advanced}\:\:{calculus}…… \\ $$$$\:\:\:\:\:\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{\mathrm{2}} }{{cosh}^{\mathrm{2}} \left({x}^{\mathrm{2}} \right)}{dx}\overset{?} {=}\frac{\sqrt{\mathrm{2}}\:−\mathrm{2}}{\mathrm{4}}\:\sqrt{\pi}\:\zeta\:\left(\:\frac{\mathrm{1}}{\mathrm{2}}\:\right)\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:………….. \\…

sin-4-x-cos-4-xdx-

Question Number 9581 by dc630248@gmail.com last updated on 18/Dec/16 $$\int\mathrm{sin}^{\mathrm{4}} {x}.\mathrm{cos}^{\mathrm{4}} {xdx} \\ $$ Answered by mrW last updated on 18/Dec/16 $$\mathrm{sin}^{\mathrm{4}} {x}.\mathrm{cos}^{\mathrm{4}} {x}=\frac{\mathrm{1}}{\mathrm{16}}\left(\mathrm{2sinxcosx}\right)^{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{16}}\mathrm{sin}^{\mathrm{4}}…