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Question Number 168341 Answers: 1 Comments: 0

∫(1+x^2 )^3 dx=?

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

Question Number 168339 Answers: 0 Comments: 0

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}

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

Question Number 168338 Answers: 1 Comments: 0

Question Number 168335 Answers: 2 Comments: 0

(x/2)+4(√(x ))+6+3+1=x

$$\frac{{x}}{\mathrm{2}}+\mathrm{4}\sqrt{{x}\:}+\mathrm{6}+\mathrm{3}+\mathrm{1}={x} \\ $$

Question Number 168325 Answers: 0 Comments: 0

2yy′′−(y′)^2 −1= ((8y^2 )/(cos^2 x))

$$\:\:\:\:\:\:\mathrm{2}{yy}''−\left({y}'\right)^{\mathrm{2}} \:−\mathrm{1}=\:\frac{\mathrm{8}{y}^{\mathrm{2}} }{\mathrm{cos}\:^{\mathrm{2}} {x}}\: \\ $$

Question Number 168324 Answers: 1 Comments: 0

Question Number 168320 Answers: 0 Comments: 0

∫(((arcsin(x))^2 )/(1+x^2 ))dx=???

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

Question Number 168311 Answers: 0 Comments: 5

Calculate∫(((arcsin(x))^2 )/(1+x^2 ))dx

$${Calculate}\int\frac{\left({arcsin}\left({x}\right)\right)^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 168303 Answers: 2 Comments: 0

calculate ∫_0 ^1 x(√(1−x^6 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}\sqrt{\mathrm{1}−{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 168297 Answers: 1 Comments: 0

prove that........ cos ((2Π)/7)+cos ((4Π)/7)+cos ((8Π)/7)=−(1/2)

$$\mathrm{prove}\:\mathrm{that}........ \\ $$$$\mathrm{cos}\:\frac{\mathrm{2}\Pi}{\mathrm{7}}+\mathrm{cos}\:\frac{\mathrm{4}\Pi}{\mathrm{7}}+\mathrm{cos}\:\frac{\mathrm{8}\Pi}{\mathrm{7}}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 168296 Answers: 2 Comments: 0

∫(5x+2)cos(2x)dx=?

$$\int\left(\mathrm{5}{x}+\mathrm{2}\right){cos}\left(\mathrm{2}{x}\right){dx}=? \\ $$

Question Number 168291 Answers: 2 Comments: 0

∫t^7 sin(t^7 )dt

$$\int{t}^{\mathrm{7}} \mathrm{sin}\left({t}^{\mathrm{7}} \right){dt} \\ $$

Question Number 168280 Answers: 1 Comments: 0

Wath is your favourite formula ???

$${Wath}\:{is}\:{your}\:{favourite}\:{formula}\:??? \\ $$

Question Number 168278 Answers: 1 Comments: 2

((log_3 (12))/(log_(36) (3)))−((log_3 (4))/(log_(108) (3))) = x x =

$$\:\:\:\:\:\frac{{log}_{\mathrm{3}} \left(\mathrm{12}\right)}{{log}_{\mathrm{36}} \left(\mathrm{3}\right)}−\frac{{log}_{\mathrm{3}} \left(\mathrm{4}\right)}{{log}_{\mathrm{108}} \left(\mathrm{3}\right)}\:=\:{x} \\ $$$$\:\:\:\:\:{x}\:=\: \\ $$

Question Number 168277 Answers: 0 Comments: 1

Prove that ((sin(((3π)/5)))/(sin(((4π)/5)))) = ((1+(√5))/2)

$$\:\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{5}}\right)}{\mathrm{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)}\:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\: \\ $$

Question Number 168276 Answers: 3 Comments: 2

y = (√(x+(√(x+(√x))))) y′ =

$$\:\:\:\:{y}\:=\:\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}} \\ $$$$\:\:\:\:{y}'\:=\: \\ $$$$\:\:\:\:\: \\ $$

Question Number 168274 Answers: 1 Comments: 0

Question Number 168267 Answers: 1 Comments: 0

Solve this integral : ∫(((√(x+1))−1)/( (√(x+1))+1))

$$\:\:\:\:\:\:{Solve}\:\:{this}\:{integral}\:: \\ $$$$\:\:\:\:\:\:\:\int\frac{\sqrt{{x}+\mathrm{1}}−\mathrm{1}}{\:\sqrt{{x}+\mathrm{1}}+\mathrm{1}} \\ $$$$ \\ $$

Question Number 168264 Answers: 1 Comments: 0

Question Number 168259 Answers: 1 Comments: 0

If the odds in favour of an event be (1/3).Find the probability of an occurrence of the event.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{odds}\:\mathrm{in}\:\mathrm{favour}\:\mathrm{of}\:\mathrm{an}\:\mathrm{event} \\ $$$$\mathrm{be}\:\frac{\mathrm{1}}{\mathrm{3}}.\mathrm{Find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of}\: \\ $$$$\mathrm{an}\:\mathrm{occurrence}\:\mathrm{of}\:\mathrm{the}\:\mathrm{event}. \\ $$

Question Number 168256 Answers: 1 Comments: 0

lim_(x→∞) ((n!)/n^n )=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{n}!}{{n}^{{n}} }=? \\ $$

Question Number 168248 Answers: 0 Comments: 1

Question Number 168247 Answers: 2 Comments: 0

Solve for x ((8^x +27^x )/(12^x +18^x ))=(7/6) Mastermind

$${Solve}\:{for}\:{x} \\ $$$$\frac{\mathrm{8}^{{x}} +\mathrm{27}^{{x}} }{\mathrm{12}^{{x}} +\mathrm{18}^{{x}} }=\frac{\mathrm{7}}{\mathrm{6}} \\ $$$$ \\ $$$${Mastermind} \\ $$

Question Number 168244 Answers: 3 Comments: 0

Question Number 168233 Answers: 1 Comments: 0

{ ((u_0 = 3 : u_1 = 4)),((u_(n+1) = u_n + 6u_(n−1) )) :} Express u_n in terms of n

$$\begin{cases}{{u}_{\mathrm{0}} \:=\:\mathrm{3}\::\:{u}_{\mathrm{1}} \:=\:\mathrm{4}}\\{{u}_{{n}+\mathrm{1}} \:=\:{u}_{{n}} \:+\:\mathrm{6}{u}_{{n}−\mathrm{1}} }\end{cases} \\ $$$${Express}\:{u}_{{n}} \:{in}\:{terms}\:{of}\:{n} \\ $$

Question Number 168231 Answers: 0 Comments: 4

∫ x (√(1−x^6 )) dx = ?

$$\int\:\:{x}\:\sqrt{\mathrm{1}−{x}^{\mathrm{6}} }\:\:{dx}\:\:=\:\:? \\ $$

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