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Question Number 72175 Answers: 1 Comments: 3

(x^2 +y^2 −1)^3 =x^2 y^3 how large (area) is your heart ♥ ?

$$\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} =\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{3}} \\ $$$$ \\ $$$${how}\:{large}\:\left({area}\right)\:{is}\:{your}\:{heart}\:\heartsuit\:? \\ $$

Question Number 72093 Answers: 1 Comments: 2

a, b, c, d, e, f, g, h, i, j are different nonnegative integers . Find all numbers that satisfy a × bc × def = ghij

$${a},\:{b},\:{c},\:{d},\:{e},\:{f},\:{g},\:{h},\:{i},\:{j}\:\:\:{are}\:\:\:{different}\:\:{nonnegative}\:\:{integers}\:. \\ $$$${Find}\:\:{all}\:\:{numbers}\:\:{that}\:\:{satisfy} \\ $$$$\:\:\:\:\:{a}\:×\:{bc}\:×\:{def}\:\:=\:\:{ghij} \\ $$

Question Number 72086 Answers: 0 Comments: 0

(sin^(−1) (x))^v +(cos^(−1) (x)^v ) for any value v what is it in terms of pi

$$\left(\mathrm{sin}^{−\mathrm{1}} \left({x}\right)\right)^{{v}} +\left(\mathrm{cos}^{−\mathrm{1}} \left({x}\right)^{{v}} \right) \\ $$$${for}\:{any}\:{value}\:{v}\:{what}\:{is}\:{it}\: \\ $$$${in}\:{terms}\:{of}\:{pi} \\ $$$$ \\ $$

Question Number 72071 Answers: 2 Comments: 1

a) lim_(x→0) ((3x−sin(2x))/(2x−sin(3x)))

$$\left.{a}\right)\underset{{x}\rightarrow\mathrm{0}} {\:{lim}}\frac{\mathrm{3}{x}−{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}{x}−{sin}\left(\mathrm{3}{x}\right)} \\ $$$$ \\ $$$$ \\ $$

Question Number 72060 Answers: 0 Comments: 2

∫((sin x)/(1+sin x+sin 2x))dx=?

$$\int\frac{\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{sin}\:{x}+\mathrm{sin}\:\mathrm{2}{x}}{dx}=? \\ $$

Question Number 72048 Answers: 1 Comments: 0

Question Number 72047 Answers: 0 Comments: 1

Question Number 72046 Answers: 0 Comments: 2

Question Number 72044 Answers: 1 Comments: 3

y=ln(3x^2 +x) solve y^((6)) (x)

$$\mathrm{y}=\mathrm{ln}\left(\mathrm{3x}^{\mathrm{2}} +\mathrm{x}\right)\:\:\:\mathrm{solve}\:\mathrm{y}^{\left(\mathrm{6}\right)} \left(\mathrm{x}\right) \\ $$

Question Number 72036 Answers: 2 Comments: 3

Question Number 72027 Answers: 1 Comments: 1

if a+b+c =3 and a^3 +b^3 +c^3 =27 find a^2 +b^2 +c^2

$${if}\:{a}+{b}+{c}\:=\mathrm{3}\:\:{and}\:{a}^{\mathrm{3}} \:+{b}^{\mathrm{3}} \:+{c}^{\mathrm{3}} =\mathrm{27}\:{find}\:{a}^{\mathrm{2}} \:+{b}^{\mathrm{2}} \:+{c}^{\mathrm{2}} \\ $$

Question Number 72026 Answers: 1 Comments: 1

prove that x^4 divide (1+x^2 )^n −nx^2 −1 without use binomial formula.

$${prove}\:{that}\:{x}^{\mathrm{4}} \:{divide}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} −{nx}^{\mathrm{2}} −\mathrm{1}\:\:{without}\:{use}\:\:{binomial} \\ $$$${formula}. \\ $$

Question Number 72025 Answers: 1 Comments: 1

calculate ∫_0 ^∞ e^(−αx) ln(x)dx with α>0

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\alpha{x}} {ln}\left({x}\right){dx}\:\:{with}\:\alpha>\mathrm{0} \\ $$

Question Number 72024 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^n (1−(x/n))^n ln(1+x)dx

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\int_{\mathrm{0}} ^{{n}} \left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {ln}\left(\mathrm{1}+{x}\right){dx} \\ $$

Question Number 72023 Answers: 1 Comments: 1

find lim_(n→+∞) Σ_(1≤i<j≤n) (1/((ij)^2 ))

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{\mathrm{1}}{\left({ij}\right)^{\mathrm{2}} } \\ $$

Question Number 72021 Answers: 0 Comments: 2

calculate interms of n U_n =Σ_(1≤i<j≤n) sin(((iπ)/n))sin(((jπ)/n))

$${calculate}\:{interms}\:{of}\:{n}\:\:{U}_{{n}} =\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:{sin}\left(\frac{{i}\pi}{{n}}\right){sin}\left(\frac{{j}\pi}{{n}}\right) \\ $$

Question Number 72020 Answers: 1 Comments: 1

calculate Σ_(k=0) ^n C_n ^k cos(((kπ)/(2n)))

$${calculate}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}}\right) \\ $$

Question Number 72018 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (((−1)^n )/(n^3 (n+1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 72017 Answers: 0 Comments: 0

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

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

Question Number 72016 Answers: 1 Comments: 2

find U_n =∫_0 ^∞ ((cos(nx))/((3+nx^2 )^2 ))dx with n integr

$${find}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({nx}\right)}{\left(\mathrm{3}+{nx}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{with}\:{n}\:{integr} \\ $$

Question Number 72015 Answers: 0 Comments: 1

find ∫ ((arctan(x−(1/x)))/(x^2 +1))dx

$${find}\:\int\:\:\frac{{arctan}\left({x}−\frac{\mathrm{1}}{{x}}\right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 72012 Answers: 0 Comments: 1

find f(α) =∫_0 ^∞ ((arctan(αx))/((x^2 +3)^2 ))dx with α real.

$${find}\:{f}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx}\:\:{with}\:\alpha\:{real}. \\ $$

Question Number 72011 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((ln(3+x^2 ))/((x^2 +1)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{3}+{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 71993 Answers: 1 Comments: 0

Question Number 71986 Answers: 1 Comments: 0

Question Number 71984 Answers: 1 Comments: 0

x, y ∈ I ⊂ R prove that ∣cosx − siny∣ ≤ ∣x − y∣

$$\:{x},\:{y}\:\in\:{I}\:\subset\:\mathbb{R}\:\:{prove}\:\:{that}\:\:\mid{cosx}\:−\:{siny}\mid\:\leqslant\:\mid{x}\:−\:{y}\mid \\ $$

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