Category: Integration

Question Number 219060 by MrGaster last updated on 19/Apr/25 $${\int }_0^{\mathrm{\infty }}\frac{{\sin }^m}{x^n}dx,n\in \mathbb{N},m\in \mathbb{N},n⩽m$$ Answered by Nicholas666 last updated on 19/Apr/25 $$\frac{\pi}{\mathrm{2}} \…

Question Number 219078 by zetamaths last updated on 19/Apr/25 $$(1-\frac{1}{2m}).\underset{k=1}{\sum ^m}{(-1)}^{k-1}\cdot k\cdot \frac{{(m!)}^2}{(m-k)!(m+k)!}=\frac{1}{4}Proofthisformula$$ Answered by MrGaster last updated on 19/Apr/25 $$\mathrm{Let}\:{S}\left({m}\right)\underset{{k}=\mathrm{1}} {\overset{{m}}…

Question Number 218896 by Nicholas666 last updated on 17/Apr/25 $$$$$$\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e};$$$$\:\mid\int\int\int_{\left[\mathrm{0},\infty\right]^{\mathrm{3}} } \boldsymbol{{f}}\frac{\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{x}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{y}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{z}}\right)}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} }\mid\leqslant\boldsymbol{{C}}\left(\int\int\int_{\mathbb{R}_{+} ^{\mathrm{3}} } \mid\boldsymbol{{f}}\mid\left(\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}}…

Question Number 218949 by Spillover last updated on 17/Apr/25 Commented by Spillover last updated on 17/Apr/25 $$ans=\frac{\ln (1+\sqrt{2)}}{\sqrt{2}}$$ Commented by Nicholas666 last updated on…

Question Number 218951 by Spillover last updated on 17/Apr/25 Commented by Spillover last updated on 17/Apr/25 $$ans=\frac{z}{\sinh z}$$ Answered by Spillover last updated on…