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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)

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

Question Number 143147 Answers: 0 Comments: 1

montrer que lasuite U_n =(H_n /n^2 ) est bornee H_n =Σ_(k=1) ^n (1/n^2 )

$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{lasuite}\:\mathrm{U}_{\mathrm{n}} =\frac{\mathrm{H}_{\mathrm{n}} }{\mathrm{n}^{\mathrm{2}} }\:\mathrm{est}\:\mathrm{bornee} \\ $$$$\mathrm{H}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 143142 Answers: 1 Comments: 0

∫((x^2 −1)/(x^2 +1))∙(1/( (√(1+x^4 ))))dx

$$\int\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\centerdot\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }}\mathrm{dx} \\ $$

Question Number 143139 Answers: 0 Comments: 0

Let a,b,c > 0 and a+b+c = 3. Prove that (1+a^2 )(1+b^2 )(1+c^2 ) ≤ (1+(1/( (√(abc)))))^3

$$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}\:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{b}^{\mathrm{2}} \right)\left(\mathrm{1}+{c}^{\mathrm{2}} \right)\:\leqslant\:\left(\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt{{abc}}}\right)^{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 143138 Answers: 0 Comments: 0

Let a,b > 0 and a+b = 2. Prove that (1+a^2 )(1+b^2 ) ≤ (1+(1/( (√(ab)))))^2

$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}\:=\:\mathrm{2}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{b}^{\mathrm{2}} \right)\:\leqslant\:\left(\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt{{ab}}}\right)^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 143134 Answers: 1 Comments: 0

Question Number 143131 Answers: 2 Comments: 0

find the partial sums of Σ_(n=1) ^∞ (1/(n^2 (n+1)))

$${find}\:{the}\:{partial}\:{sums}\:{of}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)} \\ $$

Question Number 143130 Answers: 1 Comments: 1

Question Number 143127 Answers: 2 Comments: 0

Question Number 143122 Answers: 0 Comments: 0

Let a,b ∈[0,1] and a+b ≤ 1. Prove that (1/(1+a))+(1/(1+b))+(1/2) ≤ (2/(a+b))

$$\mathrm{Let}\:{a},{b}\:\in\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{and}\:{a}+{b}\:\leqslant\:\mathrm{1}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{a}}+\frac{\mathrm{1}}{\mathrm{1}+{b}}+\frac{\mathrm{1}}{\mathrm{2}}\:\leqslant\:\frac{\mathrm{2}}{{a}+{b}}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 143114 Answers: 1 Comments: 0

solve: lim_(z→i) ((3z^4 −2z^3 +8z^2 −2z+5)/(z−i))=?

$${solve}: \\ $$$$\underset{{z}\rightarrow{i}} {\mathrm{lim}}\frac{\mathrm{3}{z}^{\mathrm{4}} −\mathrm{2}{z}^{\mathrm{3}} +\mathrm{8}{z}^{\mathrm{2}} −\mathrm{2}{z}+\mathrm{5}}{{z}−{i}}=? \\ $$

Question Number 143113 Answers: 0 Comments: 1

x^3 =(1/(3!))∫_0 ^x f(x−t)f(t)dt f(x)=?

$$\mathrm{x}^{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{3}!}\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{f}\left(\mathrm{x}−\mathrm{t}\right)\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 143109 Answers: 3 Comments: 1

Question Number 143105 Answers: 0 Comments: 1

((cos(3x))/(sin(2x))) = 0

$$\frac{{cos}\left(\mathrm{3}{x}\right)}{{sin}\left(\mathrm{2}{x}\right)}\:=\:\mathrm{0} \\ $$

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