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Question Number 151983 Answers: 1 Comments: 0

∫_(−∞) ^(+∞) (((1−ix)/(1+ix)))^n (((1+ix)/(1−ix)))^m (1/(1+x^2 ))dx

$$\int_{−\infty} ^{+\infty} \left(\frac{\mathrm{1}−{ix}}{\mathrm{1}+{ix}}\right)^{{n}} \left(\frac{\mathrm{1}+{ix}}{\mathrm{1}−{ix}}\right)^{{m}} \frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 151979 Answers: 3 Comments: 0

lim_(x→0) (((1+mx)^n - (1+nx)^m )/x^2 ) = ? ; m;n∈N

$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{1}+\mathrm{mx}\right)^{\boldsymbol{\mathrm{n}}} \:-\:\left(\mathrm{1}+\mathrm{nx}\right)^{\boldsymbol{\mathrm{m}}} }{\mathrm{x}^{\mathrm{2}} }\:=\:?\:\:;\:\:\mathrm{m};\mathrm{n}\in\mathbb{N} \\ $$

Question Number 151973 Answers: 1 Comments: 1

Question Number 152026 Answers: 0 Comments: 1

Question Number 151965 Answers: 2 Comments: 0

x , y ∈ R & sin(x )+ cos (y ) =1 then max ( sin(y) + cos (x) ) =? ....

$$ \\ $$$$\:\:\:{x}\:,\:{y}\:\in\:\mathbb{R}\: \\ $$$$\:\:\:\:\:\:\&\:{sin}\left({x}\:\right)+\:{cos}\:\left({y}\:\right)\:=\mathrm{1} \\ $$$${then}\:\:{max}\:\left(\:{sin}\left({y}\right)\:+\:{cos}\:\left({x}\right)\:\right)\:=? \\ $$$$\:\:.... \\ $$

Question Number 151961 Answers: 1 Comments: 1

Question Number 151959 Answers: 1 Comments: 0

Question Number 151958 Answers: 0 Comments: 0

f ( x ) = a −(√((x/(1+x)) )) , D_( f) : [ 0, ∞) , a≥ 1 , h (x ):=(√(( f^( −1) (a−ax ))/(f^( −1) ( a− 2x )))) D_( h) = ? ( D := Domain )

$${f}\:\left(\:{x}\:\right)\:=\:{a}\:−\sqrt{\frac{{x}}{\mathrm{1}+{x}}\:}\:\:\:,\:{D}_{\:{f}} \::\:\left[\:\mathrm{0},\:\infty\right) \\ $$$$,\:{a}\geqslant\:\mathrm{1}\:\:\:,\:\:{h}\:\left({x}\:\right):=\sqrt{\frac{\:{f}^{\:−\mathrm{1}} \left({a}−{ax}\:\right)}{{f}^{\:−\mathrm{1}} \left(\:{a}−\:\mathrm{2}{x}\:\right)}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{D}_{\:{h}} \:=\:?\:\:\:\left(\:\:\:{D}\::=\:{Domain}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 151956 Answers: 0 Comments: 0

lim_(n→∞) Σ_(k=1) ^n arctan (1/(2k^2 )) arctan ((2k^2 - 1)/(2k^2 )) = ?

$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{arctan}\:\frac{\mathrm{1}}{\mathrm{2k}^{\mathrm{2}} }\:\mathrm{arctan}\:\frac{\mathrm{2k}^{\mathrm{2}} \:-\:\mathrm{1}}{\mathrm{2k}^{\mathrm{2}} }\:=\:? \\ $$

Question Number 151954 Answers: 1 Comments: 0

nice...calculus 𝛗 := ∫_0 ^( (π/2)) x^( 3) . cot (x )dx =(a/(16)) a :=? m.n...

$$ \\ $$$$\:\:\:\:{nice}...{calculus} \\ $$$$\: \\ $$$$\:\:\:\:\:\boldsymbol{\phi}\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}^{\:\mathrm{3}} .\:{cot}\:\left({x}\:\right){dx}\:=\frac{{a}}{\mathrm{16}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\::=? \\ $$$${m}.{n}... \\ $$

Question Number 151951 Answers: 4 Comments: 1

nice ... mathematics S:= Σ_(n=1) ^∞ (( ζ (2n ))/(n . 16^( n) )) = ? ......■

$$ \\ $$$$\:\:\:\:\:{nice}\:...\:{mathematics} \\ $$$$\:\:\:\:\:\:\mathrm{S}:=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\zeta\:\left(\mathrm{2}{n}\:\right)}{{n}\:.\:\mathrm{16}^{\:{n}} }\:=\:?\:......\blacksquare \\ $$$$ \\ $$

Question Number 151948 Answers: 0 Comments: 0

Question Number 151943 Answers: 1 Comments: 1

Prove arctan1+arctan2+arctan3=π

$$\mathrm{Prove}\:\mathrm{arctan1}+\mathrm{arctan2}+\mathrm{arctan3}=\pi \\ $$

Question Number 151942 Answers: 0 Comments: 0

If x , y > 1 : Min ( (( x^( 4) )/((y −1 )^( 2) )) + (( y^( 4) )/(( x − 1 )^( 2) )) ) = ? ......■ ? m.n...

$$ \\ $$$$\:\:\:\mathrm{If}\:\:{x}\:,\:{y}\:>\:\mathrm{1}\:\:\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Min}\:\left(\:\frac{\:{x}^{\:\mathrm{4}} }{\left({y}\:−\mathrm{1}\:\right)^{\:\mathrm{2}} }\:+\:\frac{\:{y}^{\:\mathrm{4}} }{\left(\:{x}\:−\:\mathrm{1}\:\right)^{\:\mathrm{2}} }\:\right)\:=\:?\:......\blacksquare\:\:?\:\: \\ $$$$\:\:\:\:{m}.{n}... \\ $$$$ \\ $$

Question Number 151941 Answers: 1 Comments: 0

Find solution set of equation 6cos x−8sin x=5(√3) 0°≤x≤360°

$$\mathrm{Find}\:\mathrm{solution}\:\mathrm{set}\:\mathrm{of}\:\mathrm{equation} \\ $$$$\:\mathrm{6cos}\:\mathrm{x}−\mathrm{8sin}\:\mathrm{x}=\mathrm{5}\sqrt{\mathrm{3}} \\ $$$$\:\mathrm{0}°\leqslant\mathrm{x}\leqslant\mathrm{360}° \\ $$

Question Number 151940 Answers: 1 Comments: 0

Question Number 151934 Answers: 1 Comments: 0

Question Number 151922 Answers: 0 Comments: 0

((cos ((2π)/9)))^(1/3) +((cos ((4π)/9)))^(1/3) −((cos (π/9)))^(1/3) =?

$$\:\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\:+\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\mathrm{4}\pi}{\mathrm{9}}}−\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\pi}{\mathrm{9}}}\:=? \\ $$$$ \\ $$

Question Number 151921 Answers: 0 Comments: 0

Question Number 152980 Answers: 2 Comments: 10

Question Number 151911 Answers: 0 Comments: 10

Determine the digit a and prime numbers x;y;z such that x<y, z<1000 and x + y^(2a) = z

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{digit}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\mathrm{prime} \\ $$$$\mathrm{numbers}\:\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}};\boldsymbol{\mathrm{z}}\:\mathrm{such}\:\mathrm{that}\:\boldsymbol{\mathrm{x}}<\boldsymbol{\mathrm{y}},\:\boldsymbol{\mathrm{z}}<\mathrm{1000} \\ $$$$\mathrm{and}\:\:\mathrm{x}\:+\:\mathrm{y}^{\mathrm{2}\boldsymbol{\mathrm{a}}} \:=\:\mathrm{z} \\ $$

Question Number 151907 Answers: 1 Comments: 2

If the area of a convex quadrilateral is 2k^2 and the sum of its diagonals is 4k^2 , then show that this quadrilateral is an orthodiagonal one.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{is}\:\mathrm{2}\boldsymbol{\mathrm{k}}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{diagonals} \\ $$$$\mathrm{is}\:\mathrm{4}\boldsymbol{\mathrm{k}}^{\mathrm{2}} ,\:\mathrm{then}\:\mathrm{show}\:\mathrm{that}\:\mathrm{this}\:\mathrm{quadrilateral} \\ $$$$\mathrm{is}\:\mathrm{an}\:\mathrm{orthodiagonal}\:\mathrm{one}. \\ $$

Question Number 151905 Answers: 1 Comments: 0

Question Number 151899 Answers: 1 Comments: 0

Question Number 151897 Answers: 1 Comments: 0

Question Number 151895 Answers: 0 Comments: 0

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