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AllQuestion and Answers: Page 611

Question Number 155560 Answers: 0 Comments: 0

Question Number 155553 Answers: 0 Comments: 0

∫_(−∞) ^( ∞) ((sin(x^2 )cos(x^3 ))/((ln((sin(x)cos(x))^2 ))^2 +1)) dx

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

Question Number 155547 Answers: 1 Comments: 0

let a;b;c>0 and a+b+c=3 find min value of the expression: S = abc + (a-1)^2 + (b-1)^2 + (c-1)^2

$$\mathrm{let}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3} \\ $$$$\mathrm{find}\:\mathrm{min}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}: \\ $$$$\mathrm{S}\:=\:\mathrm{abc}\:+\:\left(\mathrm{a}-\mathrm{1}\right)^{\mathrm{2}} \:+\:\left(\mathrm{b}-\mathrm{1}\right)^{\mathrm{2}} \:+\:\left(\mathrm{c}-\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 155545 Answers: 1 Comments: 6

Solve in R ((5x)/( (√(5x^2 + 4)) + 7(√x))) + ((x + 2)/( (√(x^2 - 3x - 18)) + 2(√x)))

$$\mathrm{Solve}\:\mathrm{in}\:\mathbb{R} \\ $$$$\frac{\mathrm{5x}}{\:\sqrt{\mathrm{5x}^{\mathrm{2}} \:+\:\mathrm{4}}\:+\:\mathrm{7}\sqrt{\mathrm{x}}}\:+\:\frac{\mathrm{x}\:+\:\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{3x}\:-\:\mathrm{18}}\:+\:\mathrm{2}\sqrt{\mathrm{x}}} \\ $$

Question Number 155542 Answers: 1 Comments: 4

find the tylor series expantion of ((z^2 −1)/((z+1)(z+3)))

$${find}\:{the}\:{tylor}\:{series}\:{expantion}\:{of}\:\frac{{z}^{\mathrm{2}} −\mathrm{1}}{\left({z}+\mathrm{1}\right)\left({z}+\mathrm{3}\right)} \\ $$

Question Number 155541 Answers: 0 Comments: 0

form a partial equation from (x^2 /2)+(y^2 /2)=z^2

$${form}\:{a}\:{partial}\:{equation}\:{from}\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{y}^{\mathrm{2}} }{\mathrm{2}}={z}^{\mathrm{2}} \\ $$

Question Number 155540 Answers: 1 Comments: 0

f(5−4x)=27x^(2000) −2187x^(204) −x^4 +3x+8 f(3x−7)=?

$$\:\:\mathrm{f}\left(\mathrm{5}−\mathrm{4x}\right)=\mathrm{27x}^{\mathrm{2000}} −\mathrm{2187x}^{\mathrm{204}} −\mathrm{x}^{\mathrm{4}} +\mathrm{3x}+\mathrm{8} \\ $$$$\mathrm{f}\left(\mathrm{3x}−\mathrm{7}\right)=? \\ $$

Question Number 155537 Answers: 1 Comments: 0

Question Number 155533 Answers: 1 Comments: 0

coefficiient of x^( 60) = ? P = (x−1)(x^( 2) −1)(x^( 3) −1)...(x^( 15) −1)

$$ \\ $$$$\:\:\:{coefficiient}\:{of}\:\:\:\:{x}^{\:\mathrm{60}} \:=\:? \\ $$$$ \\ $$$$\:\:\:\mathrm{P}\:=\:\left({x}−\mathrm{1}\right)\left({x}^{\:\mathrm{2}} −\mathrm{1}\right)\left({x}^{\:\mathrm{3}} −\mathrm{1}\right)...\left({x}^{\:\mathrm{15}} −\mathrm{1}\right) \\ $$$$ \\ $$

Question Number 155531 Answers: 0 Comments: 4

Question Number 155524 Answers: 0 Comments: 1

how can convert the interval of the intigral ∫_0 ^( 1) f(x) dx to the interval ∫_0 ^( ∞) f(x) dx ?

$$\boldsymbol{{how}}\:\boldsymbol{{can}}\:\boldsymbol{{convert}}\:\boldsymbol{{the}}\:\boldsymbol{{interval}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{intigral}}\: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:\boldsymbol{{to}}\:\boldsymbol{{the}}\:\boldsymbol{{interval}}\:\int_{\mathrm{0}} ^{\:\infty} \:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:? \\ $$

Question Number 155527 Answers: 2 Comments: 0

If f(tan^2 (θ/2))= (2/(1+cos θ)) , find f(sin (θ/2)).

$$\mathrm{If}\:{f}\left(\mathrm{tan}^{\mathrm{2}} \:\frac{\theta}{\mathrm{2}}\right)=\:\frac{\mathrm{2}}{\mathrm{1}+\mathrm{cos}\:\theta}\:,\:\mathrm{find}\:{f}\left(\mathrm{sin}\:\frac{\theta}{\mathrm{2}}\right). \\ $$

Question Number 155510 Answers: 1 Comments: 0

Question Number 155506 Answers: 1 Comments: 0

Question Number 155500 Answers: 2 Comments: 0

Question Number 155497 Answers: 1 Comments: 0

Question Number 155496 Answers: 1 Comments: 0

Given I_n =∫_(nπ) ^((n+1)π) e^(−x) sinx dx , n∈N. 1. Find a relation between I_(n+1) and I_n .

$${Given}\:{I}_{{n}} =\underset{{n}\pi} {\overset{\left({n}+\mathrm{1}\right)\pi} {\int}}{e}^{−{x}} {sinx}\:{dx}\:,\:{n}\in\mathbb{N}. \\ $$$$\mathrm{1}.\:{Find}\:{a}\:{relation}\:{between}\:{I}_{{n}+\mathrm{1}} {and}\:{I}_{{n}} . \\ $$

Question Number 155495 Answers: 1 Comments: 0

if a;b;c;d∈R verify a+2b+3c+4d=6 then find min(a^2 +b^2 +c^2 +d^2 )

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c};\mathrm{d}\in\mathbb{R}\:\:\mathrm{verify}\:\:\mathrm{a}+\mathrm{2b}+\mathrm{3c}+\mathrm{4d}=\mathrm{6} \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\boldsymbol{\mathrm{min}}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \right) \\ $$

Question Number 155488 Answers: 1 Comments: 0

Question Number 155481 Answers: 1 Comments: 0

if a;b;c∈[1;∞) then prove that a^(1/a) ; b^(1/b) ; c^(1/c) are the sides of a triangle.

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\in\left[\mathrm{1};\infty\right) \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\:\mathrm{a}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}} \:;\:\mathrm{b}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}}} \:;\:\mathrm{c}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{c}}}} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}. \\ $$

Question Number 155480 Answers: 1 Comments: 0

Find the positive integer solution of the equation: x^3 + y^3 = 911(xy + 49)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{solution} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:=\:\mathrm{911}\left(\mathrm{xy}\:+\:\mathrm{49}\right) \\ $$

Question Number 155479 Answers: 1 Comments: 5

Find: 𝛀 =∫_( 0) ^( ∞) (((√x) ln(x))/(x^2 + 1)) dx = ?

$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\sqrt{\mathrm{x}}\:\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 155478 Answers: 1 Comments: 0

Π_(k = 1 ) ^n (1 + (1/k))^k = V_n find V_n

$$\underset{{k}\:=\:\mathrm{1}\:} {\overset{{n}} {\prod}}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{k}}\right)^{{k}} \:=\:{V}_{{n}} \\ $$$${find}\:{V}_{{n}} \\ $$

Question Number 155477 Answers: 1 Comments: 0

Question Number 155512 Answers: 1 Comments: 0

𝛀 =∫_( 0) ^( 1) ((tan^(-1) x - tan^(-1) ((1/x)))/(1 + x^2 )) ∙ ((1+x)/(1-x)) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{tan}^{-\mathrm{1}} \boldsymbol{\mathrm{x}}\:-\:\mathrm{tan}^{-\mathrm{1}} \left(\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)}{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} }\:\centerdot\:\frac{\mathrm{1}+\mathrm{x}}{\mathrm{1}-\mathrm{x}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 155466 Answers: 2 Comments: 2

tan^2 ((π/(16)))+tan^2 (((2π)/(16)))+…+tan^2 (((7π)/(16)))=?

$$\:\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\pi}{\mathrm{16}}\right)+\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\mathrm{2}\pi}{\mathrm{16}}\right)+\ldots+\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\mathrm{7}\pi}{\mathrm{16}}\right)=? \\ $$

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