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91x^2 +84y^2 −24xy+406x−392y+799=0 find the eccentricity,focus,length of major & minor axis,directrix & length of eccentric perpendicular

$$\mathrm{91}{x}^{\mathrm{2}} +\mathrm{84}{y}^{\mathrm{2}} −\mathrm{24}{xy}+\mathrm{406}{x}−\mathrm{392}{y}+\mathrm{799}=\mathrm{0} \\ $$$${find}\:{the}\:{eccentricity},{focus},{length}\:{of}\:{major}\:\&\:{minor}\:{axis},{directrix}\:\&\:{length}\:{of}\:{eccentric}\:{perpendicular} \\ $$$$ \\ $$

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let U_n =∫_0 ^1 (x^n )(√(1−x^(2n+1) )))dx 1) find a equivalent of U_n (n∼∞) 2) study the comvergence of Σ U_n

$$\left.{let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}^{{n}} \right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}{n}+\mathrm{1}} }\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{U}_{{n}} \left({n}\sim\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{comvergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

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∫(1+x^2 )^3 dx=?

$$\int\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} {dx}=? \\ $$

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A point in rectangular coordinates (x,y,z) can be represented in spherical coordinates (r,θ,ϕ) by: x = r sin θ sin ϕ, y = sin θ sin ϕ, z = sin ϕ, 0 ≤ θ ≤ 2π , 0 ≤ ϕ ≤ π (a) Calculate the Jacobian of the transformation ((∂(x,y,z))/(∂(r,θ,ϕ))) (b) Calculate the volume of the region delimited by the sphere: S = {x,y,z ∈R^3 , x^2 +y^2 +z^2 ≤ R^2 , R>0}

$$\mathrm{A}\:\mathrm{point}\:\mathrm{in}\:\mathrm{rectangular}\:\mathrm{coordinates}\: \\ $$$$\left({x},{y},{z}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{represented}\:\mathrm{in}\:\mathrm{spherical} \\ $$$$\mathrm{coordinates}\:\left({r},\theta,\varphi\right)\:\mathrm{by}: \\ $$$$\:{x}\:=\:{r}\:\mathrm{sin}\:\theta\:\mathrm{sin}\:\varphi,\:{y}\:=\:\mathrm{sin}\:\theta\:\mathrm{sin}\:\varphi,\: \\ $$$${z}\:=\:\mathrm{sin}\:\varphi,\:\mathrm{0}\:\leqslant\:\theta\:\leqslant\:\mathrm{2}\pi\:,\:\mathrm{0}\:\leqslant\:\varphi\:\leqslant\:\pi \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{Jacobian}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{transformation}\:\frac{\partial\left({x},{y},{z}\right)}{\partial\left({r},\theta,\varphi\right)} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region} \\ $$$$\mathrm{delimited}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sphere}: \\ $$$$\:\:{S}\:=\:\left\{{x},{y},{z}\:\in\mathbb{R}^{\mathrm{3}} \:,\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:\leqslant\:{R}^{\mathrm{2}} ,\:{R}>\mathrm{0}\right\} \\ $$

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