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Question Number 78284 Answers: 0 Comments: 1

calculate ∫∫_D xy(√(x^2 +2y^2 ))dxdy D={(x,y)/0≤x≤1 and 0≤y≤(√(1−x^2 ))}

$${calculate}\:\int\int_{{D}} {xy}\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} }{dxdy} \\ $$$${D}=\left\{\left({x},{y}\right)/\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right\} \\ $$

Question Number 78283 Answers: 0 Comments: 3

calculate ∫∫_D (x^2 +2y)dxdy with D={(x,y)∈R^2 / x^2 ≥y and y≥x^2 }

$${calculate}\:\int\int_{{D}} \:\left({x}^{\mathrm{2}} +\mathrm{2}{y}\right){dxdy} \\ $$$${with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:{x}^{\mathrm{2}} \geqslant{y}\:{and}\:{y}\geqslant{x}^{\mathrm{2}} \right\} \\ $$

Question Number 78281 Answers: 0 Comments: 0

calculate ∫∫_W ((x^2 −3y^2 )/e^(x^2 +y^2 ) )dxdy with W =[0,1]×[0,1]

$${calculate}\:\:\int\int_{{W}} \:\frac{{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} }{{e}^{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } }{dxdy} \\ $$$${with}\:{W}\:=\left[\mathrm{0},\mathrm{1}\right]×\left[\mathrm{0},\mathrm{1}\right] \\ $$

Question Number 78280 Answers: 0 Comments: 0

find by recurrence J_(n,p) =∫_0 ^1 x^n (arctanx)^p dx stydy the serie Σ_(n≥0 and p≥0) J_(n,p)

$${find}\:{by}\:{recurrence} \\ $$$${J}_{{n},{p}} \:=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} \left({arctanx}\right)^{{p}} {dx} \\ $$$${stydy}\:{the}\:{serie}\:\sum_{{n}\geqslant\mathrm{0}\:{and}\:{p}\geqslant\mathrm{0}} \:\:{J}_{{n},{p}} \\ $$

Question Number 78277 Answers: 0 Comments: 0

calculate ∫∫_W (e^(−x^2 −y^2 ) /(2(√(x^2 +y^2 ))+3))dxdy with W ={ (x,y)/ x>0 and y>0}

$${calculate}\:\int\int_{{W}} \:\:\:\frac{{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } }{\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }+\mathrm{3}}{dxdy} \\ $$$${with}\:{W}\:=\left\{\:\left({x},{y}\right)/\:{x}>\mathrm{0}\:{and}\:{y}>\mathrm{0}\right\} \\ $$

Question Number 78276 Answers: 0 Comments: 1

find I_n =∫∫_([1,n]^2 ) (√(x^2 +y^2 ))ln(x^2 +y^2 )dxdy

$${find}\:{I}_{{n}} =\int\int_{\left[\mathrm{1},{n}\right]^{\mathrm{2}} } \:\:\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{ln}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dxdy} \\ $$

Question Number 78273 Answers: 0 Comments: 1

let f(θ) =∫_0 ^(π/4) (dx/(1+sinθ sinx)) with 0<θ<(π/2) 1) explicite f(θ) 2) calculate ∫_0 ^(π/4) (dx/((1+sinθ sinx)^2 ))

$${let}\:{f}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\mathrm{1}+{sin}\theta\:{sinx}} \\ $$$$ \\ $$$${with}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{explicite}\:{f}\left(\theta\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{\left(\mathrm{1}+{sin}\theta\:{sinx}\right)^{\mathrm{2}} } \\ $$

Question Number 78271 Answers: 0 Comments: 1

calculate A_θ =∫_0 ^(π/2) (dx/(2+cosθ sinx)) −π<θ<π

$${calculate}\:{A}_{\theta} \:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{2}+{cos}\theta\:{sinx}} \\ $$$$−\pi<\theta<\pi \\ $$

Question Number 78270 Answers: 1 Comments: 1

find ∫_0 ^1 ((ln(1−x^2 )ln(x))/x^2 )dx prove first the convergence.

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){ln}\left({x}\right)}{{x}^{\mathrm{2}} }{dx} \\ $$$${prove}\:{first}\:{the}\:{convergence}. \\ $$

Question Number 78269 Answers: 1 Comments: 1

let f(a) =∫_0 ^∞ ((cos(ax))/(x^2 +a^2 ))dx with a>0 find ∫_1 ^2 f(a)da

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({ax}\right)}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}\:{with} \\ $$$${a}>\mathrm{0}\:\:{find}\:\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {f}\left({a}\right){da} \\ $$

Question Number 78267 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) ((x^2 −x)/(x^4 −x^2 +3))dx

$${find}\:\:\int_{−\infty} ^{+\infty} \:\frac{{x}^{\mathrm{2}} −{x}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

Question Number 78266 Answers: 1 Comments: 2

calculate f(a)=∫_0 ^1 ln(1−ax^3 )dx with 0<a<1

$${calculate}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{ax}^{\mathrm{3}} \right){dx} \\ $$$${with}\:\mathrm{0}<{a}<\mathrm{1} \\ $$

Question Number 78264 Answers: 0 Comments: 4

let I =∫_0 ^π x cos^4 x dxand J=∫_0 ^π x sin^4 xdx 1) calculate I+J and I−J 2) find the values of I and J

$${let}\:{I}\:=\int_{\mathrm{0}} ^{\pi} {x}\:{cos}^{\mathrm{4}} {x}\:{dxand}\:{J}=\int_{\mathrm{0}} ^{\pi} {x}\:{sin}^{\mathrm{4}} {xdx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}+{J}\:{and}\:{I}−{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:{I}\:{and}\:{J} \\ $$

Question Number 78263 Answers: 1 Comments: 2

find lim_(n→+∞) Σ_(k=1) ^n sin((1/(k+n)))

$${find}\:{lim}_{{n}\rightarrow+\infty} \sum_{{k}=\mathrm{1}} ^{{n}} {sin}\left(\frac{\mathrm{1}}{{k}+{n}}\right) \\ $$

Question Number 78265 Answers: 1 Comments: 0

find ∫ ((sin^3 x)/(tan^5 x))dx

$${find}\:\int\:\:\frac{{sin}^{\mathrm{3}} {x}}{{tan}^{\mathrm{5}} {x}}{dx} \\ $$

Question Number 78256 Answers: 0 Comments: 0

Evaluate Σa_1 a_2 a_3 as a function of a_i

$$\mathrm{Evaluate}\:\Sigma\mathrm{a}_{\mathrm{1}} \mathrm{a}_{\mathrm{2}} \mathrm{a}_{\mathrm{3}} \: \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:\:\mathrm{a}_{\mathrm{i}} \: \\ $$$$ \\ $$$$ \\ $$

Question Number 78261 Answers: 0 Comments: 3

let U_n =∫_0 ^1 (x^n /(1+x))dx calculate U_n +U_(n+1)

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\:\:{calculate} \\ $$$${U}_{{n}} \:+{U}_{{n}+\mathrm{1}} \\ $$

Question Number 78251 Answers: 1 Comments: 0

∫((x+4)/(x−(x)^(1/3) )) dx

$$\int\frac{{x}+\mathrm{4}}{{x}−\sqrt[{\mathrm{3}}]{{x}}}\:{dx}\: \\ $$

Question Number 78246 Answers: 0 Comments: 0

common equation of conic sections ax^2 +bxy+cy^2 +dx+ey+f=0 if b≠0 we rotate tan 2α =(b/(a−c)) [if a=c ⇒ α=45°] { ((x=x′cos α −y′sin α)),((y=x′sin α +y′cos α)) :} we now have [using x, y again instead of x′, y′] Ax^2 +Cy^2 +Dx+Ey+F=0 now complete the squares A(x+(D/(2A)))^2 +C(y+(E/(2C)))^2 +(F−(D^2 /(4A^2 ))−(E^2 /(4C^2 )))=0 { ((x=x′−(D/(2A)))),((y=y′−(E/(2C)))) :} we now have [using x, y again instead of x′, y′] one of these { ((Ax^2 +By^2 +C=0)),((Ax+By^2 +C=0)),((Ax^2 +By+C=0)) :} a, c, A, C, A, B, C ≠0 in all other cases use your brain to interprete which kind of curve (if any) we have

$$\mathrm{common}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{conic}\:\mathrm{sections} \\ $$$${ax}^{\mathrm{2}} +{bxy}+{cy}^{\mathrm{2}} +{dx}+{ey}+{f}=\mathrm{0} \\ $$$$\mathrm{if}\:{b}\neq\mathrm{0}\:\mathrm{we}\:\mathrm{rotate} \\ $$$$\mathrm{tan}\:\mathrm{2}\alpha\:=\frac{{b}}{{a}−{c}}\:\left[\mathrm{if}\:{a}={c}\:\Rightarrow\:\alpha=\mathrm{45}°\right] \\ $$$$\begin{cases}{{x}={x}'\mathrm{cos}\:\alpha\:−{y}'\mathrm{sin}\:\alpha}\\{{y}={x}'\mathrm{sin}\:\alpha\:+{y}'\mathrm{cos}\:\alpha}\end{cases} \\ $$$$\mathrm{we}\:\mathrm{now}\:\mathrm{have}\:\left[\mathrm{using}\:{x},\:{y}\:\mathrm{again}\:\mathrm{instead}\:\mathrm{of}\:{x}',\:{y}'\right] \\ $$$${Ax}^{\mathrm{2}} +{Cy}^{\mathrm{2}} +{Dx}+{Ey}+{F}=\mathrm{0} \\ $$$$\mathrm{now}\:\mathrm{complete}\:\mathrm{the}\:\mathrm{squares} \\ $$$${A}\left({x}+\frac{{D}}{\mathrm{2}{A}}\right)^{\mathrm{2}} +{C}\left({y}+\frac{{E}}{\mathrm{2}{C}}\right)^{\mathrm{2}} +\left({F}−\frac{{D}^{\mathrm{2}} }{\mathrm{4}{A}^{\mathrm{2}} }−\frac{{E}^{\mathrm{2}} }{\mathrm{4}{C}^{\mathrm{2}} }\right)=\mathrm{0} \\ $$$$\begin{cases}{{x}={x}'−\frac{{D}}{\mathrm{2}{A}}}\\{{y}={y}'−\frac{{E}}{\mathrm{2}{C}}}\end{cases} \\ $$$$\mathrm{we}\:\mathrm{now}\:\mathrm{have}\:\left[\mathrm{using}\:{x},\:{y}\:\mathrm{again}\:\mathrm{instead}\:\mathrm{of}\:{x}',\:{y}'\right] \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{these} \\ $$$$\begin{cases}{\mathcal{A}{x}^{\mathrm{2}} +\mathcal{B}{y}^{\mathrm{2}} +\mathcal{C}=\mathrm{0}}\\{\mathcal{A}{x}+\mathcal{B}{y}^{\mathrm{2}} +\mathcal{C}=\mathrm{0}}\\{\mathcal{A}{x}^{\mathrm{2}} +\mathcal{B}{y}+\mathcal{C}=\mathrm{0}}\end{cases} \\ $$$$ \\ $$$${a},\:{c},\:{A},\:{C},\:\mathcal{A},\:\mathcal{B},\:\mathcal{C}\:\neq\mathrm{0} \\ $$$$\mathrm{in}\:\mathrm{all}\:\mathrm{other}\:\mathrm{cases}\:\mathrm{use}\:\mathrm{your}\:\mathrm{brain}\:\mathrm{to}\:\mathrm{interprete} \\ $$$$\mathrm{which}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{curve}\:\left(\mathrm{if}\:\mathrm{any}\right)\:\mathrm{we}\:\mathrm{have} \\ $$

Question Number 78233 Answers: 0 Comments: 8

given 5x+22y=18 find for x,y integer

$${given}\:\mathrm{5}{x}+\mathrm{22}{y}=\mathrm{18} \\ $$$${find}\:{for}\:{x},{y}\:{integer} \\ $$

Question Number 78216 Answers: 1 Comments: 5

Question Number 78207 Answers: 1 Comments: 6

what minimum value of f(x)= 9tan^2 (x)+4cot^2 (x)

$${what}\:{minimum}\: \\ $$$${value}\:{of}\:{f}\left({x}\right)=\:\mathrm{9tan}\:^{\mathrm{2}} \left({x}\right)+\mathrm{4cot}\:^{\mathrm{2}} \left({x}\right) \\ $$

Question Number 78198 Answers: 1 Comments: 4

Question Number 78177 Answers: 1 Comments: 10

what equation of ellips with F_1 (1,2) F_2 (3,4) and a = (√3)

$${what}\:{equation}\:{of}\:{ellips} \\ $$$${with}\:{F}_{\mathrm{1}} \left(\mathrm{1},\mathrm{2}\right)\:{F}_{\mathrm{2}} \left(\mathrm{3},\mathrm{4}\right)\:{and}\:{a}\:=\:\sqrt{\mathrm{3}} \\ $$

Question Number 78168 Answers: 0 Comments: 6

Solve for x, y, z if: x^3 + y^3 + z^3 = 42

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x},\:\mathrm{y},\:\mathrm{z}\:\:\mathrm{if}:\:\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \:\:=\:\:\mathrm{42} \\ $$

Question Number 78163 Answers: 0 Comments: 2

∫ (√(1 + 3 sin(θ) + sin^2 (θ))) dθ

$$\int\:\sqrt{\mathrm{1}\:+\:\mathrm{3}\:\mathrm{sin}\left(\theta\right)\:+\:\mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\:\:\mathrm{d}\theta \\ $$

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