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Question Number 219077    Answers: 0   Comments: 0

∫_0 ^(+∞) (((sin(n))/n))^m dn=π∙(m/2^m )∙Σ_(φ=0) ^(m/2) (−1)^∅ ∙(((n−2φ)^(m−1) )/((m−φ)!∙φ!)) Proof this formula

$$\int_{\mathrm{0}} ^{+\infty} \left(\frac{{sin}\left({n}\right)}{{n}}\right)^{{m}} {dn}=\pi\centerdot\frac{{m}}{\mathrm{2}^{{m}} }\centerdot\underset{\phi=\mathrm{0}} {\overset{{m}/\mathrm{2}} {\sum}}\left(−\mathrm{1}\right)^{\emptyset} \centerdot\frac{\left({n}−\mathrm{2}\phi\right)^{{m}−\mathrm{1}} }{\left({m}−\phi\right)!\centerdot\phi!}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Proof}\:{this}\:{formula} \\ $$

Question Number 219076    Answers: 0   Comments: 0

Question Number 219071    Answers: 3   Comments: 0

Question Number 219070    Answers: 0   Comments: 0

Question Number 219069    Answers: 0   Comments: 0

Question Number 219068    Answers: 2   Comments: 0

Question Number 219067    Answers: 2   Comments: 1

Question Number 219066    Answers: 4   Comments: 0

Question Number 219065    Answers: 3   Comments: 0

Question Number 219060    Answers: 2   Comments: 0

∫_0 ^∞ ((sin^m )/x^n )dx,n∈N,m∈N,n≤m

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{{m}} }{{x}^{{n}} }{dx},{n}\in\mathbb{N},{m}\in\mathbb{N},{n}\leqslant{m} \\ $$

Question Number 219135    Answers: 1   Comments: 0

Question Number 219025    Answers: 3   Comments: 0

Question Number 219004    Answers: 0   Comments: 0

Question Number 219003    Answers: 1   Comments: 0

Question Number 222659    Answers: 1   Comments: 0

Question Number 218970    Answers: 4   Comments: 0

2,12,18,48,50,.....

$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{2},\mathrm{12},\mathrm{18},\mathrm{48},\mathrm{50},..... \\ $$$$ \\ $$

Question Number 218957    Answers: 4   Comments: 0

Question Number 218956    Answers: 2   Comments: 0

Question Number 218955    Answers: 4   Comments: 0

Question Number 218954    Answers: 2   Comments: 1

Question Number 218953    Answers: 4   Comments: 0

Question Number 218952    Answers: 4   Comments: 7

Question Number 218951    Answers: 2   Comments: 1

Question Number 218950    Answers: 7   Comments: 1

Question Number 218949    Answers: 0   Comments: 1

Question Number 218896    Answers: 0   Comments: 1

prove; ∣∫∫∫_([0,∞]^3 ) f((J_0 (x)J_0 (y)J_0 (z))/(1+x^2 y^2 z^2 ))∣≤C(∫∫∫_R_+ ^3 ∣f∣(1+x^2 y^2 z^2 )^2 )^(1/2)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{prove}}; \\ $$$$\:\mid\int\int\int_{\left[\mathrm{0},\infty\right]^{\mathrm{3}} } \boldsymbol{{f}}\frac{\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{x}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{y}}\right)\boldsymbol{{J}}_{\mathrm{0}} \left(\boldsymbol{{z}}\right)}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} }\mid\leqslant\boldsymbol{{C}}\left(\int\int\int_{\mathbb{R}_{+} ^{\mathrm{3}} } \mid\boldsymbol{{f}}\mid\left(\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} \right)^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:\:\:\:\:\:\: \\ $$$$ \\ $$

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