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AllQuestion and Answers: Page 1304

Question Number 83188 Answers: 1 Comments: 0

If Σ_(a = 0) ^(n−1) (2a+1)x^2 +(n^2 +4n−5)x+16 = 0 is a perfect square such that n ∈ Z^+ . what is the value of x +n ?

$$\mathrm{If}\:\underset{\mathrm{a}\:=\:\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\left(\mathrm{2a}+\mathrm{1}\right)\mathrm{x}^{\mathrm{2}} +\left(\mathrm{n}^{\mathrm{2}} +\mathrm{4n}−\mathrm{5}\right)\mathrm{x}+\mathrm{16} \\ $$$$=\:\mathrm{0}\:\mathrm{is}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\mathrm{n}\:\in\:\mathbb{Z}^{+} \:.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{x}\:+\mathrm{n}\:?\: \\ $$

Question Number 83166 Answers: 2 Comments: 2

what is coefficient x^(5 ) in expansion (1+x^2 )^5 ×(1+x)^4

$$\mathrm{what}\:\mathrm{is}\:\mathrm{coefficient}\:\mathrm{x}^{\mathrm{5}\:} \mathrm{in}\:\mathrm{expansion} \\ $$$$\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{5}} ×\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{4}} \: \\ $$

Question Number 83159 Answers: 1 Comments: 5

3x (xy−2)dx + (x^3 +2y) dy =0 find the solution

$$\mathrm{3x}\:\left(\mathrm{xy}−\mathrm{2}\right)\mathrm{dx}\:+\:\left(\mathrm{x}^{\mathrm{3}} +\mathrm{2y}\right)\:\mathrm{dy}\:=\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution} \\ $$

Question Number 83156 Answers: 1 Comments: 2

find the derivtive of y=e^(cos x)

$${find}\:{the}\:{derivtive}\:{of}\:{y}={e}^{\mathrm{cos}\:{x}} \\ $$

Question Number 83149 Answers: 0 Comments: 3

y=e^(tant )

$${y}={e}^{\mathrm{tan}{t}\:} \\ $$

Question Number 83164 Answers: 2 Comments: 2

Evaluate: ∫_0 ^( (π/2)) (( 1)/(1+cos 𝛂 cos x))dx

$$ \\ $$$$ \\ $$$$\:\mathrm{Evaluate}: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\:\:\mathrm{1}}{\mathrm{1}+\boldsymbol{\mathrm{cos}}\:\boldsymbol{\alpha}\:\boldsymbol{\mathrm{cos}}\:\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}} \\ $$

Question Number 83146 Answers: 2 Comments: 0

∫_( 0) ^(π/2) ((√(cot x))/((√(cot x)) + (√(tan x)))) dx =

$$\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\:\frac{\sqrt{\mathrm{cot}\:{x}}}{\sqrt{\mathrm{cot}\:{x}}\:+\:\sqrt{\mathrm{tan}\:{x}}}\:{dx}\:= \\ $$

Question Number 83145 Answers: 1 Comments: 1

∫_( 1) ^4 e^(√x) dx =

$$\:\underset{\:\mathrm{1}} {\overset{\mathrm{4}} {\int}}\:{e}^{\sqrt{{x}}} \:{dx}\:= \\ $$

Question Number 83144 Answers: 1 Comments: 0

show that Σ_(n=1) ^∞ (1/((2n−1)(3n−1)))=(1/6)((√3) π−9log(3)+log(4096))

$${show}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{3}{n}−\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{6}}\left(\sqrt{\mathrm{3}}\:\pi−\mathrm{9}{log}\left(\mathrm{3}\right)+{log}\left(\mathrm{4096}\right)\right) \\ $$

Question Number 83139 Answers: 0 Comments: 0

let c_0 >0 and ∀ n∈N c_(n+1) =(√((1/2)(c_n +(1/c_n ) ) )) Explicit c_n in term of n and c_0

$${let}\:\:\:{c}_{\mathrm{0}} \:>\mathrm{0}\:\:{and}\:\:\forall\:{n}\in\mathbb{N}\:\:{c}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left({c}_{{n}} +\frac{\mathrm{1}}{{c}_{{n}} }\:\right)\:}\:\:\: \\ $$$${Explicit}\:\:{c}_{{n}} \:{in}\:{term}\:{of}\:{n}\:{and}\:\:{c}_{\mathrm{0}} \:\: \\ $$

Question Number 83131 Answers: 0 Comments: 3

y=x^(4ln x)

$${y}=\boldsymbol{{x}}^{\mathrm{4ln}\:\boldsymbol{{x}}} \\ $$

Question Number 83127 Answers: 1 Comments: 0

what is the range of x(√3) +y if x^2 +y^2 −xy= 3 ?

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{x}\sqrt{\mathrm{3}}\:+\mathrm{y}\: \\ $$$$\mathrm{if}\:\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} −\mathrm{xy}=\:\mathrm{3}\:? \\ $$

Question Number 83123 Answers: 0 Comments: 6

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

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

Question Number 83115 Answers: 1 Comments: 1

∫cos xe^(sin x) dx

$$\int\mathrm{cos}\:{xe}^{\mathrm{sin}\:{x}} {dx} \\ $$

Question Number 83110 Answers: 0 Comments: 10

bounded by the curve y=(√(4-x)) y=0 y=1

$${bounded}\:{by}\:{the}\:{curve}\:{y}=\sqrt{\mathrm{4}-{x}}\:{y}=\mathrm{0}\:{y}=\mathrm{1} \\ $$

Question Number 83109 Answers: 0 Comments: 1

∫_(1/e) ^e (dt/t)

$$\int_{\mathrm{1}/\boldsymbol{{e}}} ^{{e}} \frac{\boldsymbol{{dt}}}{\boldsymbol{{t}}} \\ $$

Question Number 83108 Answers: 1 Comments: 0

prove that ∫_0 ^(π/4) ((cos(nx))/(cos^n (x))) dx =2^n [(π/8)−Σ_(k=1) ^(n−1) ((sin(((kπ)/4)))/(2k((√2))^k ))] n∈N^∗

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{cos}\left({nx}\right)}{{cos}^{{n}} \left({x}\right)}\:{dx}\:=\mathrm{2}^{{n}} \left[\frac{\pi}{\mathrm{8}}−\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{{sin}\left(\frac{{k}\pi}{\mathrm{4}}\right)}{\mathrm{2}{k}\left(\sqrt{\mathrm{2}}\right)^{{k}} }\right]\:{n}\in{N}^{\ast} \\ $$

Question Number 83104 Answers: 1 Comments: 0

∫((e^x dx)/(3+e^x ))

$$\int\frac{{e}^{{x}} {dx}}{\mathrm{3}+{e}^{{x}} } \\ $$

Question Number 83102 Answers: 0 Comments: 3

Question Number 83096 Answers: 0 Comments: 1

∫tan x^4 dx

$$\int\mathrm{tan}\:{x}^{\mathrm{4}} {dx} \\ $$

Question Number 83095 Answers: 0 Comments: 0

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83094 Answers: 0 Comments: 0

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83093 Answers: 0 Comments: 1

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83092 Answers: 0 Comments: 0

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83085 Answers: 1 Comments: 2

1) find ∫ (dx/((x^2 +1)^4 )) 2)calculate ∫_0 ^∞ (dx/((x^2 +1)^4 ))

$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 83078 Answers: 0 Comments: 1

If I_1 =∫_e ^e^2 (dx/(log x)) and I_2 = ∫_( 1) ^2 (e^x /x) dx, then

$$\mathrm{If}\:\:\:{I}_{\mathrm{1}} =\underset{{e}} {\overset{{e}^{\mathrm{2}} } {\int}}\:\frac{{dx}}{\mathrm{log}\:{x}}\:\:\mathrm{and}\:\:{I}_{\mathrm{2}} =\:\underset{\:\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\frac{{e}^{{x}} }{{x}}\:{dx},\:\mathrm{then} \\ $$

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