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AllQuestion and Answers: Page 1403

Question Number 61646 Answers: 0 Comments: 0

let f(x) =e^(−ax) arctan(3x) with a>0 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f (x) at integr serie . 3) calculate ∫_0 ^∞ f(x)dx .

$${let}\:{f}\left({x}\right)\:={e}^{−{ax}} \:{arctan}\left(\mathrm{3}{x}\right)\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:\left({x}\right)\:{at}\:{integr}\:{serie}\:. \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:{f}\left({x}\right){dx}\:. \\ $$

Question Number 61645 Answers: 0 Comments: 1

calculate ∫∫_D ∫(√(x^2 +y^2 +z^2 ))dxdydz with D ={(x,y,z) / 0≤x≤1 ,1≤y≤2 , 2≤z≤3 }

$${calculate}\:\int\int_{{D}} \int\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }{dxdydz} \\ $$$${with}\:{D}\:=\left\{\left({x},{y},{z}\right)\:/\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\:,\mathrm{1}\leqslant{y}\leqslant\mathrm{2}\:\:,\:\mathrm{2}\leqslant{z}\leqslant\mathrm{3}\:\right\} \\ $$

Question Number 61652 Answers: 1 Comments: 1

solve inside C z^4 =((1−i)/(1+i(√3)))

$${solve}\:{inside}\:{C}\:\:{z}^{\mathrm{4}} \:=\frac{\mathrm{1}−{i}}{\mathrm{1}+{i}\sqrt{\mathrm{3}}} \\ $$

Question Number 61651 Answers: 0 Comments: 4

let p(x) =(x+i(√3))^n +(x−i(√3))^n with x real 1) simlify p(x) 2) find the roots of P(x) 3)decompose inside C[x] p(x) 4) calculate ∫_0 ^1 p(x)dx

$${let}\:{p}\left({x}\right)\:=\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} +\left({x}−{i}\sqrt{\mathrm{3}}\right)^{{n}} \:\:\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{simlify}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){decompose}\:{inside}\:{C}\left[{x}\right]\:\:{p}\left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {p}\left({x}\right){dx}\: \\ $$

Question Number 61650 Answers: 0 Comments: 4

solve inside N^2 (x+1)(y+2) =2xy

$${solve}\:{inside}\:{N}^{\mathrm{2}} \:\:\:\:\left({x}+\mathrm{1}\right)\left({y}+\mathrm{2}\right)\:=\mathrm{2}{xy} \\ $$

Question Number 61648 Answers: 0 Comments: 1

calculate ∫∫_W (x^2 −2y^2 )(√(x^2 +y^2 +3))dxdy with W ={ (x,y) ∈ R^2 / 1≤x ≤(√3) and x^2 +y^2 −2y ≤ 2 }

$${calculate}\:\int\int_{{W}} \:\left({x}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{3}}{dxdy}\:\:\:\:{with} \\ $$$${W}\:=\left\{\:\left({x},{y}\right)\:\in\:{R}^{\mathrm{2}} \:\:/\:\:\:\:\mathrm{1}\leqslant{x}\:\leqslant\sqrt{\mathrm{3}}\:\:{and}\:\:\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} −\mathrm{2}{y}\:\leqslant\:\mathrm{2}\:\right\} \\ $$

Question Number 61635 Answers: 1 Comments: 0

2(∫_0 ^( x) y^3 cos xdx)[((yd^2 y)/dx^2 )−((dy/dx))^2 ] = ky^5 sin x ; y(0)=a, y′(0)=0 . solve the differential equation. (Laplace tranforms might be helpful, i think).

$$\mathrm{2}\left(\int_{\mathrm{0}} ^{\:{x}} {y}^{\mathrm{3}} \mathrm{cos}\:{xdx}\right)\left[\frac{{yd}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} \right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:{ky}^{\mathrm{5}} \mathrm{sin}\:{x}\:\:\:\:\:;\:\: \\ $$$$\:\:{y}\left(\mathrm{0}\right)={a},\:{y}'\left(\mathrm{0}\right)=\mathrm{0}\:. \\ $$$$\:{solve}\:{the}\:{differential}\:{equation}. \\ $$$$\left({Laplace}\:{tranforms}\:{might}\right. \\ $$$$\left.\:\:\:\:{be}\:{helpful},\:{i}\:{think}\right). \\ $$

Question Number 61622 Answers: 0 Comments: 0

If (√(x (√((x+1) (√((x+2) (√((x+3) (√(...)))))))))) = 2019 so (√(x + (√((x+1) + (√((x+2) + (√((x+3) + (√(...)))))))))) = ?

$${If}\:\:\:\sqrt{{x}\:\sqrt{\left({x}+\mathrm{1}\right)\:\sqrt{\left({x}+\mathrm{2}\right)\:\sqrt{\left({x}+\mathrm{3}\right)\:\sqrt{...}}}}}\:\:\:=\:\:\:\mathrm{2019} \\ $$$${so}\:\:\sqrt{{x}\:+\:\sqrt{\left({x}+\mathrm{1}\right)\:+\:\:\sqrt{\left({x}+\mathrm{2}\right)\:+\:\:\sqrt{\left({x}+\mathrm{3}\right)\:+\:\:\sqrt{...}}}}}\:\:\:=\:\:\:? \\ $$$$ \\ $$

Question Number 61667 Answers: 1 Comments: 0

∫(√(tan(x))) dx

$$\int\sqrt{{tan}\left({x}\right)}\:{dx}\: \\ $$

Question Number 61625 Answers: 1 Comments: 0

1+(1/(1+(1/(1+(1/(1+(1/(1+(1/(1+...))))))))))=

$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+...}}}}}= \\ $$

Question Number 61614 Answers: 0 Comments: 1

∫_(−1) ^1 (d/dx) (tan^(−1) (1/x))dx =

$$\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\:\frac{{d}}{{dx}}\:\left(\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{{x}}\right){dx}\:= \\ $$

Question Number 61613 Answers: 1 Comments: 1

S = 1 + (3/(2018)) + (5/(2018^2 )) + (7/(2018^3 )) + ... 4S − S^2 = ?

$${S}\:\:=\:\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{2018}}\:+\:\frac{\mathrm{5}}{\mathrm{2018}^{\mathrm{2}} }\:+\:\frac{\mathrm{7}}{\mathrm{2018}^{\mathrm{3}} }\:+\:... \\ $$$$\mathrm{4}{S}\:−\:{S}^{\mathrm{2}} \:\:=\:\:? \\ $$

Question Number 61605 Answers: 0 Comments: 7

solve at Z^2 2x +5y =4

$${solve}\:\:{at}\:{Z}^{\mathrm{2}} \:\:\:\:\mathrm{2}{x}\:+\mathrm{5}{y}\:=\mathrm{4} \\ $$

Question Number 61601 Answers: 1 Comments: 1

calvulate ∫∫_w (x^2 −y^2 )e^(−x−y) dxdy with W={(x,y)∈R^2 /0≤x≤1 and 1≤y≤3}

$${calvulate}\:\int\int_{{w}} \left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right){e}^{−{x}−{y}} {dxdy} \\ $$$${with}\:{W}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\right. \\ $$$$\left.\mathrm{1}\leqslant{y}\leqslant\mathrm{3}\right\} \\ $$

Question Number 61591 Answers: 1 Comments: 8

solve for z∈C (z)^(1/2) =−1 (z)^(1/3) =−1 (z)^(1/4) =−1

$$\mathrm{solve}\:\mathrm{for}\:{z}\in\mathbb{C} \\ $$$$\sqrt[{\mathrm{2}}]{{z}}=−\mathrm{1} \\ $$$$\sqrt[{\mathrm{3}}]{{z}}=−\mathrm{1} \\ $$$$\sqrt[{\mathrm{4}}]{{z}}=−\mathrm{1} \\ $$

Question Number 61569 Answers: 1 Comments: 3

Question Number 61566 Answers: 1 Comments: 0

∫_2 ^4 ((√(ln(9−(6−x)))/((√(ln(9−x))) + (√(ln(3−x))))) dx

$$\int_{\mathrm{2}} ^{\mathrm{4}} \:\frac{\sqrt{{ln}\left(\mathrm{9}−\left(\mathrm{6}−{x}\right)\right.}}{\sqrt{{ln}\left(\mathrm{9}−{x}\right)}\:+\:\sqrt{{ln}\left(\mathrm{3}−{x}\right)}}\:{dx} \\ $$

Question Number 61559 Answers: 2 Comments: 1

Question Number 61554 Answers: 1 Comments: 1

Question Number 61545 Answers: 0 Comments: 0

Question Number 61537 Answers: 2 Comments: 2

Question Number 61536 Answers: 0 Comments: 1

1)let U_n =Σ_(k=0) ^n (−1)^k =1−1+1−1+...(n+1 terms) is lim_(n→+∞) U_n exist ? find U_n by using integr part[..] 2) let V_n = Σ_(k=1) ^n k(−1)^k = −1+2 −3+4+.....(nterms) is lim_(n→+∞) V_n exist find V_n by using integr part[..]

$$\left.\mathrm{1}\right){let}\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:=\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+...\left({n}+\mathrm{1}\:{terms}\right) \\ $$$${is}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} {exist}\:?\:\:{find}\:{U}_{{n}} \:{by}\:{using}\:{integr}\:{part}\left[..\right] \\ $$$$\left.\mathrm{2}\right)\:{let}\:{V}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} \:\:=\:−\mathrm{1}+\mathrm{2}\:−\mathrm{3}+\mathrm{4}+.....\left({nterms}\right) \\ $$$${is}\:{lim}_{{n}\rightarrow+\infty} {V}_{{n}} \:{exist}\: \\ $$$${find}\:{V}_{{n}} {by}\:{using}\:{integr}\:{part}\left[..\right] \\ $$

Question Number 61535 Answers: 0 Comments: 0

calculate ∫_0 ^(π/2) ((ln(1+cosx))/(cosx)) dx

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

Question Number 61534 Answers: 0 Comments: 0

calculate f(a) =∫∫_W (x+ay)e^(−x) e^(−ay) dxdy with W_a ={(x,y)∈R^2 /x≥0 ,y≥0 , x+ay ≤1 } a>0

$${calculate}\:{f}\left({a}\right)\:=\int\int_{{W}} \:\left({x}+{ay}\right){e}^{−{x}} \:{e}^{−{ay}} {dxdy}\:{with} \\ $$$${W}_{{a}} =\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\:\:\:,\:{x}+{ay}\:\leqslant\mathrm{1}\:\right\}\:\:\:{a}>\mathrm{0} \\ $$

Question Number 61533 Answers: 0 Comments: 2

∫∫_([0,1]^2 ) ((x−y)/((x^2 +3y^(2 ) +1)^2 )) dxdy

$$\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\:\:\frac{{x}−{y}}{\left({x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}\:} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{dxdy}\: \\ $$

Question Number 61532 Answers: 0 Comments: 0

prove that (((a+b)^n )/(a^n +b^n )) <2^(n−1) ∀ n>1 (n natural)

$${prove}\:{that}\:\:\:\frac{\left({a}+{b}\right)^{{n}} }{{a}^{{n}} \:+{b}^{{n}} }\:<\mathrm{2}^{{n}−\mathrm{1}} \:\:\:\:\:\forall\:{n}>\mathrm{1}\:\:\:\:\left({n}\:{natural}\right) \\ $$

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