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AllQuestion and Answers: Page 732

Question Number 143234 Answers: 1 Comments: 0

Question Number 143231 Answers: 0 Comments: 0

.....mathematical ......Analysis.... if :: 𝛗(n):=∫_0 ^( 1) x^(2n−1) log(1+x)dx then find the value of :: Θ:= Σ_(n=1) ^∞ (−1)^n 𝛗(n) .......m.n

$$.....{mathematical}\:......{Analysis}.... \\ $$$$\:{if}\:::\:\:\:\boldsymbol{\phi}\left({n}\right):=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}{n}−\mathrm{1}} {log}\left(\mathrm{1}+{x}\right){dx} \\ $$$$\:{then}\:\:{find}\:\:{the}\:{value}\:{of}\:\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Theta:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \boldsymbol{\phi}\left({n}\right)\: \\ $$$$\:\:\:\:\:\:.......{m}.{n} \\ $$

Question Number 143230 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (x^(3n+1) /(3n+1))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{x}^{\mathrm{3n}+\mathrm{1}} }{\mathrm{3n}+\mathrm{1}}=? \\ $$

Question Number 143229 Answers: 0 Comments: 0

Let a,b,c ≥ 0 and (1+a)(1+b)(1+c) = 8. Prove that (a+((2b+1)/(a+b+1)))(b+((2c+1)/(b+c+1)))(c+((2a+1)/(c+a+1))) ≥ 8

$$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+{c}\right)\:=\:\mathrm{8}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+\frac{\mathrm{2}{b}+\mathrm{1}}{{a}+{b}+\mathrm{1}}\right)\left({b}+\frac{\mathrm{2}{c}+\mathrm{1}}{{b}+{c}+\mathrm{1}}\right)\left({c}+\frac{\mathrm{2}{a}+\mathrm{1}}{{c}+{a}+\mathrm{1}}\right)\:\geqslant\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 143225 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (1/((2021^n )(n!))) = Σ_(n=0) ^∞ (1/((2021^n )(∫_0 ^( ∞) t^n .e^(−t) dt)))

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\left(\mathrm{2021}^{{n}} \right)\left({n}!\right)}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\:\frac{\mathrm{1}}{\left(\mathrm{2021}^{{n}} \right)\left(\int_{\mathrm{0}} ^{\:\infty} {t}^{{n}} .{e}^{−{t}} \:\:{dt}\right)} \\ $$

Question Number 143222 Answers: 1 Comments: 0

If z=cos θ+i sin θ, by expand (z+(1/z))^4 (z−(1/z))^4 or other method, prove 128 sin^4 θcos^4 θ=cos 8θ−4cos 4θ+3.

$$\mathrm{If}\:{z}=\mathrm{cos}\:\theta+{i}\:\mathrm{sin}\:\theta,\:\mathrm{by}\:\mathrm{expand} \\ $$$$\left({z}+\frac{\mathrm{1}}{{z}}\right)^{\mathrm{4}} \left({z}−\frac{\mathrm{1}}{{z}}\right)^{\mathrm{4}} \mathrm{or}\:\mathrm{other}\:\mathrm{method}, \\ $$$$\mathrm{prove}\:\mathrm{128}\:\mathrm{sin}^{\mathrm{4}} \theta\mathrm{cos}^{\mathrm{4}} \theta=\mathrm{cos}\:\mathrm{8}\theta−\mathrm{4cos}\:\mathrm{4}\theta+\mathrm{3}. \\ $$

Question Number 143219 Answers: 0 Comments: 1

Question Number 143210 Answers: 1 Comments: 0

∫_0 ^1 ((7^(x+1) +3^(x+1) )/(x+1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{7}^{{x}+\mathrm{1}} +\mathrm{3}^{{x}+\mathrm{1}} }{{x}+\mathrm{1}}{dx} \\ $$

Question Number 143208 Answers: 1 Comments: 0

3^x +4x−3=x^4 find x

$$\mathrm{3}^{{x}} +\mathrm{4}{x}−\mathrm{3}={x}^{\mathrm{4}} \\ $$$${find}\:{x} \\ $$

Question Number 143200 Answers: 1 Comments: 0

Question Number 143194 Answers: 1 Comments: 0

Suppose z^(50) +z^(25) +m=0, where z=((1+i)/( (√2))) find the value of m.

$$\mathrm{Suppose}\:{z}^{\mathrm{50}} +{z}^{\mathrm{25}} +{m}=\mathrm{0},\:\mathrm{where}\:{z}=\frac{\mathrm{1}+{i}}{\:\sqrt{\mathrm{2}}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}. \\ $$

Question Number 143193 Answers: 1 Comments: 0

prove that the function f(x)=x^2 ,xε[1,4] is Riemannian integral ?

$${prove}\:{that}\:{the}\:{function}\:{f}\left({x}\right)={x}^{\mathrm{2}} \:\:,{x}\varepsilon\left[\mathrm{1},\mathrm{4}\right] \\ $$$${is}\:{Riemannian}\:{integral}\:? \\ $$

Question Number 143192 Answers: 0 Comments: 0

Question Number 143191 Answers: 1 Comments: 0

Question Number 143190 Answers: 1 Comments: 2

∫_R (e^(its) /(s+3))ds

$$\int_{\mathbb{R}} \frac{\mathrm{e}^{\mathrm{its}} }{\mathrm{s}+\mathrm{3}}\mathrm{ds} \\ $$

Question Number 143186 Answers: 0 Comments: 2

Question Number 143184 Answers: 1 Comments: 0

((10)/(25))+((28)/(125))+((82)/(625))+... = ?

$$\:\:\:\:\frac{\mathrm{10}}{\mathrm{25}}+\frac{\mathrm{28}}{\mathrm{125}}+\frac{\mathrm{82}}{\mathrm{625}}+...\:=\:? \\ $$

Question Number 143178 Answers: 0 Comments: 2

∫_1 ^∞ ((x2^x +7)/(3^x +lnx+1))dx

$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{x2}^{\mathrm{x}} +\mathrm{7}}{\mathrm{3}^{\mathrm{x}} +\mathrm{lnx}+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 143535 Answers: 0 Comments: 0

Question Number 143537 Answers: 1 Comments: 0

Question Number 143168 Answers: 2 Comments: 2

Question Number 143165 Answers: 2 Comments: 4

solve the differention equation x=p^3 −p+2 since:p=y′

$${solve}\:{the}\:{differention}\:{equation} \\ $$$${x}={p}^{\mathrm{3}} −{p}+\mathrm{2}\:\:\:{since}:{p}={y}' \\ $$

Question Number 143163 Answers: 0 Comments: 0

∫_(1/x) ^x^2 (dt/( (√(1+t^3 )))) =?

$$\int_{\frac{\mathrm{1}}{{x}}} ^{{x}^{\mathrm{2}} } \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{3}} }}\:=? \\ $$

Question Number 143156 Answers: 1 Comments: 0

lim_(n→∞) (1/n)(1+2^(1/2) +......+n^(1/n) ) ?

$${lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\left(\mathrm{1}+\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} +......+{n}^{\frac{\mathrm{1}}{{n}}} \right)\:\:? \\ $$

Question Number 143150 Answers: 0 Comments: 2

If α and β are the roots of the equation determinant (((x−cos θ),(−sin θ)),((sin θ),(x−cos θ))), find the value of α^n +β^n , where n∈N.

$$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\begin{vmatrix}{{x}−\mathrm{cos}\:\theta}&{−\mathrm{sin}\:\theta}\\{\mathrm{sin}\:\theta}&{{x}−\mathrm{cos}\:\theta}\end{vmatrix}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\alpha^{{n}} +\beta^{{n}} ,\:\mathrm{where}\:{n}\in\mathbb{N}. \\ $$

Question Number 143148 Answers: 0 Comments: 0

find v_n =Σ_(k=0) ^n (1/(3k+1)) interms of H_n H_n =Σ_(k=1) ^n (1/k)

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

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