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Question Number 61804 Answers: 1 Comments: 1

calculate Σ_(n=2) ^∞ Σ_(k=2) ^∞ (1/(k^n k!))

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{k}^{{n}} \:\:{k}!} \\ $$

Question Number 61803 Answers: 0 Comments: 3

find ∫_0 ^∞ (x^2 /(e^x^2 −1))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{{e}^{{x}^{\mathrm{2}} } −\mathrm{1}}{dx} \\ $$

Question Number 61801 Answers: 0 Comments: 3

∫_(2π) ^(4π) (√(1−cos(x))) dx

$$\underset{\mathrm{2}\pi} {\overset{\mathrm{4}\pi} {\int}}\sqrt{\mathrm{1}−{cos}\left({x}\right)}\:{dx} \\ $$

Question Number 61799 Answers: 1 Comments: 0

If a = Cosα −iSinα and b = Cosβ −iSinβ Prove that (((a+b)(1−ab))/((a−b)(1+ab))) = ((Sinα+Sinβ)/(Sinα−Sinβ))

$$\mathrm{If}\:\mathrm{a}\:=\:\mathrm{Cos}\alpha\:−\mathrm{iSin}\alpha\:\mathrm{and}\:\mathrm{b}\:=\:\mathrm{Cos}\beta\:−\mathrm{iSin}\beta \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{1}−\mathrm{ab}\right)}{\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{1}+\mathrm{ab}\right)}\:=\:\frac{\mathrm{Sin}\alpha+\mathrm{Sin}\beta}{\mathrm{Sin}\alpha−\mathrm{Sin}\beta} \\ $$

Question Number 61796 Answers: 0 Comments: 0

Question Number 61791 Answers: 1 Comments: 1

lim_(x → ∞) (6^x /(2^x + 4^x )) = ∞ ?

$$\underset{{x}\:\rightarrow\:\infty} {\mathrm{lim}}\:\:\:\frac{\mathrm{6}^{{x}} }{\mathrm{2}^{{x}} \:+\:\mathrm{4}^{{x}} }\:\:\:=\:\:\:\infty\:\:\:\:\:? \\ $$

Question Number 61785 Answers: 1 Comments: 0

∫_0 ^(√(3−x^2 )) ((xy(4−x^2 −y^2 )(√(4−x^2 −y^2 ))−xy)/3) dy

$$\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{3}−{x}^{\mathrm{2}} }} {\int}}\frac{{xy}\left(\mathrm{4}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }−{xy}}{\mathrm{3}}\:{dy} \\ $$

Question Number 61782 Answers: 0 Comments: 5

Any integer(s) which fulfill n^5 − 5n^3 + 5n + 1 ∣ n! ?

$${Any}\:\:{integer}\left({s}\right)\:\:{which}\:\:{fulfill} \\ $$$$\:\:\:\:\:\:\:\:\:\:{n}^{\mathrm{5}} \:−\:\mathrm{5}{n}^{\mathrm{3}} \:+\:\mathrm{5}{n}\:+\:\mathrm{1}\:\:\mid\:\:{n}!\:\:\:? \\ $$

Question Number 61778 Answers: 1 Comments: 0

Any integer(s) which fulfill n^3 − 5n^2 + 5n + 1 ∣ n! ?

$${Any}\:\:{integer}\left({s}\right)\:\:{which}\:\:{fulfill}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{n}^{\mathrm{3}} \:−\:\mathrm{5}{n}^{\mathrm{2}} \:+\:\mathrm{5}{n}\:+\:\mathrm{1}\:\:\mid\:\:{n}!\:\:\:\:? \\ $$

Question Number 61775 Answers: 0 Comments: 1

Question Number 61770 Answers: 0 Comments: 0

Question Number 61767 Answers: 0 Comments: 0

Question Number 61762 Answers: 2 Comments: 1

Question Number 61755 Answers: 1 Comments: 2

if f(z)=Σ_(k=1) ^n a_k z^k ,a_k ,z∈C.Prove a_k =(1/(2πi))∫_(∣z∣=r ) ((f(z))/z^(k+1) )dz

$$\mathrm{if}\:{f}\left({z}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{k}} {z}^{{k}} ,{a}_{{k}} ,{z}\in\mathbb{C}.\mathrm{Prove} \\ $$$$ \\ $$$${a}_{{k}} =\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\underset{\mid{z}\mid={r}\:} {\int}\frac{{f}\left({z}\right)}{{z}^{{k}+\mathrm{1}} }{dz} \\ $$$$ \\ $$

Question Number 61754 Answers: 0 Comments: 0

Question Number 61748 Answers: 0 Comments: 1

find ∫ (dx/(sin(2x)+tan(x)))dx

$${find}\:\:\int\:\:\:\:\frac{{dx}}{{sin}\left(\mathrm{2}{x}\right)+{tan}\left({x}\right)}{dx} \\ $$

Question Number 61752 Answers: 0 Comments: 0

solve the (de) (√(2x+1))y^′ −x^3 y = xln(x)

$${solve}\:{the}\:\left({de}\right)\:\:\:\:\:\:\sqrt{\mathrm{2}{x}+\mathrm{1}}{y}^{'} \:−{x}^{\mathrm{3}} {y}\:\:=\:{xln}\left({x}\right) \\ $$

Question Number 61751 Answers: 0 Comments: 0

use newton method to solve the equation x^4 −3x−1 =0

$${use}\:{newton}\:{method}\:{to}\:{solve}\:{the}\:{equation}\:\:{x}^{\mathrm{4}} −\mathrm{3}{x}−\mathrm{1}\:=\mathrm{0} \\ $$

Question Number 61750 Answers: 0 Comments: 0

find ∫ ((x^2 −(√(x−1)))/(2(√(x^2 +3)))) dx

$$\:{find}\:\int\:\:\frac{{x}^{\mathrm{2}} −\sqrt{{x}−\mathrm{1}}}{\mathrm{2}\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}\:{dx} \\ $$

Question Number 61749 Answers: 0 Comments: 0

find ∫ (dx/(cos(2x)+tan(x)))

$${find}\:\int\:\:\frac{{dx}}{{cos}\left(\mathrm{2}{x}\right)+{tan}\left({x}\right)} \\ $$

Question Number 61744 Answers: 0 Comments: 7

3xy^2 +x^3 =9 −−−−−(1) 3x^2 y+y^3 =18−−−−(2) Find x and y

$$\mathrm{3}{xy}^{\mathrm{2}} +{x}^{\mathrm{3}} =\mathrm{9}\:−−−−−\left(\mathrm{1}\right) \\ $$$$\mathrm{3}{x}^{\mathrm{2}} {y}+{y}^{\mathrm{3}} =\mathrm{18}−−−−\left(\mathrm{2}\right) \\ $$$${Find}\:{x}\:{and}\:{y} \\ $$

Question Number 61738 Answers: 2 Comments: 0

Question Number 61735 Answers: 0 Comments: 0

∫_(−1) ^1 (((sin(x))/(sinh^(−1) (x))))(((sin^(−1) (x))/(sinh(x)))) dx =?

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

Question Number 61733 Answers: 0 Comments: 1

Find the maximum volume tetrahedron inside an ellipsoid, parameters a,b,c .

$${Find}\:{the}\:{maximum} \\ $$$${volume}\:{tetrahedron}\:{inside}\:{an} \\ $$$${ellipsoid},\:{parameters}\:\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}}\:. \\ $$

Question Number 61721 Answers: 0 Comments: 0

calculate A =∫_0 ^∞ cos(x^n )dx and B =∫_0 ^∞ sin(x^n )dx with n≥2 (n integr natural)

$${calculate}\:{A}\:=\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{{n}} \right){dx} \\ $$$${and}\:{B}\:=\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{{n}} \right){dx}\:\:{with} \\ $$$${n}\geqslant\mathrm{2}\:\:\left({n}\:{integr}\:{natural}\right) \\ $$

Question Number 61719 Answers: 0 Comments: 1

∫(dx/(2+sin(x)))

$$\int\frac{{dx}}{\mathrm{2}+{sin}\left({x}\right)} \\ $$

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