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AllQuestion and Answers: Page 1311

Question Number 83242 Answers: 0 Comments: 3

Question Number 83240 Answers: 0 Comments: 1

Question Number 83235 Answers: 0 Comments: 2

Question Number 83229 Answers: 0 Comments: 4

Question Number 83226 Answers: 1 Comments: 1

∫_( 0) ^1 (1/(x^2 +2x cos α+1)) dx =

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{2}{x}\:\mathrm{cos}\:\alpha+\mathrm{1}}\:{dx}\:= \\ $$

Question Number 83225 Answers: 0 Comments: 1

∫_( 0) ^(100) sin (x−[x])π dx =

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{100}} {\int}}\:\mathrm{sin}\:\left({x}−\left[{x}\right]\right)\pi\:{dx}\:= \\ $$

Question Number 83214 Answers: 1 Comments: 0

calculate U_n =Σ_(k=0) ^n (1/(3k+1)) interms of H_n =Σ_(k=1) ^n (1/k)

$${calculate}\:\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{1}}\:{interms}\:{of}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 83213 Answers: 0 Comments: 0

Prove that ∀ θ∈R Σ_(n=1) ^∞ 3^(n−1) sin^3 ((θ/3^n ))= ((θ−sinθ)/4)

$${Prove}\:\:{that}\:\:\forall\:\theta\in\mathbb{R}\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{3}^{{n}−\mathrm{1}} {sin}^{\mathrm{3}} \left(\frac{\theta}{\mathrm{3}^{{n}} }\right)=\:\frac{\theta−{sin}\theta}{\mathrm{4}}\: \\ $$

Question Number 83209 Answers: 1 Comments: 0

Prove that ∀ x≠(π/4) , Π_(n=0) ^∞ (((cos((x/2^n ))+cos(((π−2x)/2^(n+1) )))/2) ) = ((4cos2x)/(π(π−4x)))

$$\:\:\:{Prove}\:{that}\:\:\forall\:\:{x}\neq\frac{\pi}{\mathrm{4}}\:\:\:,\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\left(\frac{{cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)+{cos}\left(\frac{\pi−\mathrm{2}{x}}{\mathrm{2}^{{n}+\mathrm{1}} }\right)}{\mathrm{2}}\:\right)\:=\:\frac{\mathrm{4}{cos}\mathrm{2}{x}}{\pi\left(\pi−\mathrm{4}{x}\right)}\: \\ $$

Question Number 83206 Answers: 0 Comments: 2

1) find ∫_0 ^(π/4) (dx/(2+a sinx)) (areal) 2) c explicite ∫_0 ^(π/4) ((sinx)/((2+asinx)^2 ))dx

$$\left.\mathrm{1}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{dx}}{\mathrm{2}+{a}\:{sinx}}\:\:\:\:\left({areal}\right) \\ $$$$\left.\mathrm{2}\right)\:{c}\:{explicite}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{sinx}}{\left(\mathrm{2}+{asinx}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 83205 Answers: 0 Comments: 2

∫_0 ^(ln2) (1/(cosh(x + ln4)))dx =

$$\int_{\mathrm{0}} ^{{ln}\mathrm{2}} \frac{\mathrm{1}}{{cosh}\left({x}\:+\:{ln}\mathrm{4}\right)}{dx}\:= \\ $$

Question Number 83204 Answers: 0 Comments: 1

∫_(t−1) ^t ln(x!)dx=?

$$\int_{{t}−\mathrm{1}} ^{{t}} {ln}\left({x}!\right){dx}=? \\ $$

Question Number 83202 Answers: 0 Comments: 4

find the first 4 terms in the maclaurin[ series expansion for ln (1 + 3x) hence show that if x^2 and higher powers of x are negleted, then (1 + 3x)^(3/x) = e^6 (1 −9x)

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{4}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{maclaurin}\left[\right. \\ $$$$\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\mathrm{ln}\:\left(\mathrm{1}\:+\:\mathrm{3}{x}\right)\:\mathrm{hence}\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{if}\:{x}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{higher}\:\mathrm{powers}\:\mathrm{of}\:{x}\:\mathrm{are}\:\mathrm{negleted}, \\ $$$$\mathrm{then}\: \\ $$$$\:\:\:\left(\mathrm{1}\:+\:\mathrm{3}{x}\right)^{\frac{\mathrm{3}}{{x}}} \:=\:{e}^{\mathrm{6}} \left(\mathrm{1}\:−\mathrm{9}{x}\right) \\ $$

Question Number 95177 Answers: 0 Comments: 7

Question Number 83189 Answers: 0 Comments: 6

find the 3^(rd) derivative of x^5 ln(2x) using the Leibniz theorem

$$\mathrm{find}\:\mathrm{the}\:\mathrm{3}\:^{\mathrm{rd}} \:\mathrm{derivative}\:\mathrm{of}\: \\ $$$$\mathrm{x}^{\mathrm{5}} \:\mathrm{ln}\left(\mathrm{2x}\right)\:\mathrm{using}\:\mathrm{the}\:\mathrm{Leibniz}\:\mathrm{theorem} \\ $$

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)

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

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