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AllQuestion and Answers: Page 62

Question Number 219066 Answers: 4 Comments: 0

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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

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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

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Question Number 218954 Answers: 2 Comments: 1

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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)

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

Question Number 218891 Answers: 0 Comments: 0

suppose y(x) = Σ_(n=0) ^∞ a_n x^n statisfies y′′y−(y′)^2 =e^y −1with y(0)=0 and y′(0)=1. determin a_4 .

$$ \\ $$$$\:\boldsymbol{{suppose}}\:\boldsymbol{{y}}\left(\boldsymbol{{x}}\right)\:=\:\:\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\boldsymbol{{a}}_{\boldsymbol{{n}}} \boldsymbol{{x}}^{\boldsymbol{{n}}} \boldsymbol{{statisfies}}\: \\ $$$$\:\:\:\boldsymbol{{y}}''\boldsymbol{{y}}−\left(\boldsymbol{{y}}'\right)^{\mathrm{2}} =\boldsymbol{{e}}^{\boldsymbol{{y}}} −\mathrm{1}\boldsymbol{{with}}\:\boldsymbol{{y}}\left(\mathrm{0}\right)=\mathrm{0}\:\boldsymbol{{and}}\:\boldsymbol{{y}}'\left(\mathrm{0}\right)=\mathrm{1}.\:\:\:\:\: \\ $$$$\:\:\:\:\boldsymbol{{determin}}\:\boldsymbol{{a}}_{\mathrm{4}} . \\ $$$$ \\ $$

Question Number 218890 Answers: 5 Comments: 0

Question Number 218889 Answers: 5 Comments: 0

Question Number 218888 Answers: 1 Comments: 0

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Question Number 218879 Answers: 0 Comments: 0

Calculate the following integral; ∫_(−∞) ^∞ ∫_(−∞) ^∞ ∫_(−∞) ^∞ xJ_0 ((√(x^2 +y^2 )))J_1 ((√(y^2 +z^2 )))J_2 ((√(z^2 +x^2 )))e^(−(x^2 +y^2 +z^2 )) dxdydz where J_n (u) is the Bassel function of the first kind of order n

$$ \\ $$$$\:\:\:\boldsymbol{{Calculate}}\:\boldsymbol{{the}}\:\boldsymbol{{following}}\:\boldsymbol{{integral}};\:\:\:\:\:\: \\ $$$$\:\:\int_{−\infty} ^{\infty} \int_{−\infty} ^{\infty} \int_{−\infty} ^{\infty} \boldsymbol{{xJ}}_{\mathrm{0}} \left(\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} }\right)\boldsymbol{{J}}_{\mathrm{1}} \left(\sqrt{\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} }\right)\boldsymbol{{J}}_{\mathrm{2}} \left(\sqrt{\boldsymbol{{z}}^{\mathrm{2}} +\boldsymbol{{x}}^{\mathrm{2}} }\right)\boldsymbol{{e}}^{−\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} \right)} \boldsymbol{{dxdydz}}\:\:\:\:\:\:\: \\ $$$$\:\:\:\boldsymbol{{where}}\:\boldsymbol{{J}}_{\boldsymbol{{n}}} \left(\boldsymbol{{u}}\right)\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{Bassel}}\:\boldsymbol{{function}} \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{first}}\:\boldsymbol{{kind}}\:\boldsymbol{{of}}\:\boldsymbol{{order}}\:\boldsymbol{{n}} \\ $$$$ \\ $$

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