Question Number 215134 by RoseAli last updated on 29/Dec/24 Answered by Ghisom last updated on 29/Dec/24 $$\int\sqrt{{x}−\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{{x}−\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}\:\rightarrow\:{dx}=−\frac{\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}}{{t}}{dt}\right] \\ $$$$=\int\left({t}^{\mathrm{2}} −\frac{\mathrm{4}}{{t}^{\mathrm{2}}…
Question Number 215081 by MrGaster last updated on 28/Dec/24 $$ \\ $$$$\mathrm{prove}: \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\left(\mathrm{ln}\left({x}/\left(\mathrm{1}+{x}\right)\right)^{\mathrm{4}} \mathrm{ln}\left({x}^{\mathrm{3}} \left(\mathrm{1}+{x}\right)^{\mathrm{17}} \right.\right.}{\mathrm{1}+{x}}{dx}=−\mathrm{240}\zeta\left(\mathrm{3}\right)^{\mathrm{2}} \\ $$$$ \\ $$…
Question Number 215091 by mnjuly1970 last updated on 28/Dec/24 $$ \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \:\frac{\:\Gamma^{\:\mathrm{2}} \left({n}+\mathrm{1}\right)}{\Gamma\:\left(\mathrm{2}{n}+\mathrm{1}\right)}\:=\:? \\ $$$$\:\:\:\:\:\:−−− \\ $$$$\:\:\:\:\beta\:\left({p}\:,{q}\:\right)\:=\:\frac{\:\Gamma\:\left({p}\right)\Gamma\left({q}\right)}{\Gamma\left({p}+{q}\:\right)} \\ $$ Answered by MrGaster…
Question Number 215061 by mnjuly1970 last updated on 27/Dec/24 $$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \mathrm{e}^{\:\:−\:\frac{\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{sin}\left({xy}\:\right){dxdy}=? \\ $$$$ \\ $$ Answered by…
Question Number 215020 by mnjuly1970 last updated on 26/Dec/24 $$ \\ $$$$\:\:\:\:{f}:\:\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:\rightarrow\mathbb{R}\:{is}\:{given}. \\ $$$$\:\:\:\:{f}\:''\:\:\:\:{is}\:{continuous}\:. \\ $$$$\:\:\:\:{by}\:{the}\:{way}\:\:{f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right). \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\: \\ $$$$\begin{array}{|c|}{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\:{f}\:''\:\left({x}\right)\right)^{\:\mathrm{2}} {dx}\:\geqslant\:\mathrm{3}\left({f}\:'\left(\mathrm{1}\right)\right)^{\mathrm{2}}…
Question Number 215011 by universe last updated on 25/Dec/24 Answered by MathematicalUser2357 last updated on 28/Dec/24 $$\left(\mathrm{e}\right)\:\mathrm{3} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 214925 by efronzo1 last updated on 24/Dec/24 Commented by GDVilla last updated on 24/Dec/24 $$\mathrm{Why}\:\mathrm{this}\:\mathrm{question}\:\mathrm{in}\:\mathrm{arabic}\:\mathrm{translate}\:\mathrm{pls} \\ $$ Answered by TonyCWX08 last updated on…
Question Number 214917 by Spillover last updated on 23/Dec/24 Answered by cemoosky last updated on 23/Dec/24 $$\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \centerdot\centerdot\centerdot\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}_{\mathrm{1}} ^{\mathrm{3}} +{x}_{\mathrm{2}}…
Question Number 214915 by Spillover last updated on 23/Dec/24 Answered by maths2 last updated on 23/Dec/24 $${x}\rightarrow\mathrm{1}−{x};{Let}\:{A}\:{bee}\:{the}\:{integral} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\left(\mathrm{1}+{e}^{−\mathrm{2}+\mathrm{4}{x}} \right)\left(\mathrm{5}+\mathrm{2}{x}−\mathrm{2}{x}^{\mathrm{2}} \right)}\Rightarrow \\ $$$$\mathrm{2}{A}=\int_{\mathrm{0}}…
Question Number 214770 by MathematicalUser2357 last updated on 19/Dec/24 $$\int\mathrm{sin}^{\mathrm{3}} {xdx}=? \\ $$ Answered by mr W last updated on 19/Dec/24 $$=−\int\mathrm{sin}^{\mathrm{2}} \:{x}\:{d}\:\left(\mathrm{cos}\:{x}\right) \\ $$$$=−\int\left(\mathrm{1}−\mathrm{cos}^{\mathrm{2}}…