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AllQuestion and Answers: Page 663

Question Number 151226 Answers: 0 Comments: 0

Question Number 151224 Answers: 1 Comments: 0

Question Number 151221 Answers: 1 Comments: 2

∫((sin x)/(sin (x−a)))dx

$$\int\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\left(\mathrm{x}−\mathrm{a}\right)}\mathrm{dx} \\ $$

Question Number 151220 Answers: 3 Comments: 0

show that ∫_0 ^(π/2) ((cos x)/(cos x+sin x+1))dx=(1/4)(π−2ln 2)

$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}+\mathrm{1}}\mathrm{dx}=\frac{\mathrm{1}}{\mathrm{4}}\left(\pi−\mathrm{2ln}\:\mathrm{2}\right) \\ $$

Question Number 151218 Answers: 0 Comments: 2

Question Number 151216 Answers: 1 Comments: 0

Solve for real number ((x+3))^(1/7) +((6−x))^(1/7) = (9)^(1/7)

$$\:\:\:\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{number}\: \\ $$$$\:\:\sqrt[{\mathrm{7}}]{\mathrm{x}+\mathrm{3}}\:+\sqrt[{\mathrm{7}}]{\mathrm{6}−\mathrm{x}}\:=\:\sqrt[{\mathrm{7}}]{\mathrm{9}}\: \\ $$

Question Number 151215 Answers: 1 Comments: 0

if x;y;z>0 ; x+y+z=1 and λ≥(1/6) then: 𝛌 Σ ((y + z)/x) + 3 Σ yz ≥ 6𝛌 + 1

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:;\:\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{1}\:\mathrm{and}\:\lambda\geqslant\frac{\mathrm{1}}{\mathrm{6}}\:\mathrm{then}: \\ $$$$\boldsymbol{\lambda}\:\Sigma\:\frac{\mathrm{y}\:+\:\mathrm{z}}{\mathrm{x}}\:+\:\mathrm{3}\:\Sigma\:\mathrm{yz}\:\geqslant\:\mathrm{6}\boldsymbol{\lambda}\:+\:\mathrm{1} \\ $$

Question Number 151212 Answers: 1 Comments: 0

if a_1 ,a_2 ,...a_n >1 then: (√(((a_1 -1)(a_2 -1)...(a_n -1))/((a_1 +1)(a_2 +1)...(a_n +1)))) ≤ ((a_1 a_2 ...a_n )/2^n )

$$\mathrm{if}\:\:\:\mathrm{a}_{\mathrm{1}} ,\mathrm{a}_{\mathrm{2}} ,...\mathrm{a}_{\boldsymbol{\mathrm{n}}} >\mathrm{1}\:\:\mathrm{then}: \\ $$$$\sqrt{\frac{\left(\mathrm{a}_{\mathrm{1}} -\mathrm{1}\right)\left(\mathrm{a}_{\mathrm{2}} -\mathrm{1}\right)...\left(\mathrm{a}_{\boldsymbol{\mathrm{n}}} -\mathrm{1}\right)}{\left(\mathrm{a}_{\mathrm{1}} +\mathrm{1}\right)\left(\mathrm{a}_{\mathrm{2}} +\mathrm{1}\right)...\left(\mathrm{a}_{\boldsymbol{\mathrm{n}}} +\mathrm{1}\right)}}\:\leqslant\:\frac{\mathrm{a}_{\mathrm{1}} \mathrm{a}_{\mathrm{2}} ...\mathrm{a}_{\boldsymbol{\mathrm{n}}} }{\mathrm{2}^{\boldsymbol{\mathrm{n}}} } \\ $$

Question Number 151211 Answers: 1 Comments: 0

Find the coefficient of x^9 from expression (1+x)(1+2x^2 )(1+3x^3 )(1+4x^4 )(1+5x^5 )...(1+10x^(10) )

$$\:{Find}\:{the}\:{coefficient}\:{of}\:{x}^{\mathrm{9}} \: \\ $$$${from}\:{expression}\: \\ $$$$\:\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{3}} \right)\left(\mathrm{1}+\mathrm{4}{x}^{\mathrm{4}} \right)\left(\mathrm{1}+\mathrm{5}{x}^{\mathrm{5}} \right)...\left(\mathrm{1}+\mathrm{10}{x}^{\mathrm{10}} \right) \\ $$

Question Number 151208 Answers: 0 Comments: 0

Question Number 151204 Answers: 1 Comments: 0

Question Number 151205 Answers: 1 Comments: 0

determinant ((((2+(√3))^x +1 =(2(√(2+(√3))))^x )),((x =? )))

$$\underbrace{ }\:\begin{array}{|c|c|}{\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)^{{x}} +\mathrm{1}\:=\left(\mathrm{2}\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\right)^{{x}} }\\{{x}\:=?\:}\\\hline\end{array} \\ $$

Question Number 151198 Answers: 1 Comments: 0

If a,b∈R satisfy a^4 +b^4 −6a^2 b^2 =9 and ab(a−b)(a+b)=−11 then a^2 +b^2 =?

$$\mathrm{If}\:{a},\mathrm{b}\in\mathrm{R}\:\mathrm{satisfy}\:{a}^{\mathrm{4}} +{b}^{\mathrm{4}} −\mathrm{6}{a}^{\mathrm{2}} {b}^{\mathrm{2}} =\mathrm{9}\:{and} \\ $$$${ab}\left({a}−{b}\right)\left({a}+{b}\right)=−\mathrm{11}\:\mathrm{then}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} =? \\ $$

Question Number 151193 Answers: 0 Comments: 0

Question Number 151192 Answers: 0 Comments: 1

Question Number 151189 Answers: 0 Comments: 0

In △ABC the following relationship holds: (𝛟-golden ratio) sinA + ((sinB)/𝛟) + ((sinC)/𝛟) < (1/𝛟) + ((1+(√𝛟)+𝛟)/(2𝛟))

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:\:\mathrm{the}\:\mathrm{following}\:\mathrm{relationship} \\ $$$$\mathrm{holds}:\:\left(\boldsymbol{\varphi}-\mathrm{golden}\:\mathrm{ratio}\right) \\ $$$$\mathrm{sinA}\:+\:\frac{\mathrm{sinB}}{\boldsymbol{\varphi}}\:+\:\frac{\mathrm{sinC}}{\boldsymbol{\varphi}}\:<\:\frac{\mathrm{1}}{\boldsymbol{\varphi}}\:+\:\frac{\mathrm{1}+\sqrt{\boldsymbol{\varphi}}+\boldsymbol{\varphi}}{\mathrm{2}\boldsymbol{\varphi}} \\ $$

Question Number 151182 Answers: 0 Comments: 0

∫_0 ^( ∞) ((sin(2x)ln(x))/x) dx= m.∫_0 ^( ∞) (( ln(1+2x+x^2 ))/(x(ln^2 (x)+ π^( 2) ))) dx m=?....

$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{2}{x}\right){ln}\left({x}\right)}{{x}}\:{dx}=\:{m}.\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\left(\mathrm{1}+\mathrm{2}{x}+{x}^{\mathrm{2}} \right)}{{x}\left({ln}^{\mathrm{2}} \left({x}\right)+\:\pi^{\:\mathrm{2}} \right)}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:{m}=?.... \\ $$

Question Number 151181 Answers: 2 Comments: 0

if ∣x∣<1 find x−4x^2 +9x^3 −16x^4 +...

$$\mathrm{if}\:\:\mid\boldsymbol{\mathrm{x}}\mid<\mathrm{1} \\ $$$$\mathrm{find}\:\:\mathrm{x}−\mathrm{4x}^{\mathrm{2}} +\mathrm{9x}^{\mathrm{3}} −\mathrm{16x}^{\mathrm{4}} +... \\ $$

Question Number 151179 Answers: 2 Comments: 0

if ∣x∣<1 find x+2x^2 +3x^3 +...

$$\mathrm{if}\:\:\mid\boldsymbol{\mathrm{x}}\mid<\mathrm{1} \\ $$$$\mathrm{find}\:\:\mathrm{x}+\mathrm{2x}^{\mathrm{2}} +\mathrm{3x}^{\mathrm{3}} +... \\ $$

Question Number 151175 Answers: 0 Comments: 0

Question Number 151174 Answers: 2 Comments: 0

((a+(√(3∙((a+(√(3∙((a+(√(3∙...))))^(1/3) ))))^(1/3) ))))^(1/3) = 3 find a=?

$$\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}\centerdot\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}\centerdot\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}\centerdot...}}}}}}\:\:=\:\mathrm{3}\: \\ $$$$\mathrm{find}\:\:{a}=? \\ $$

Question Number 151191 Answers: 0 Comments: 0

Question Number 151197 Answers: 1 Comments: 0

etude de la monotonie? svp u_n =Σ_(k=1) ^n (1/k)−ln(n)

$${etude}\:{de}\:{la}\:{monotonie}?\:{svp} \\ $$$${u}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}−{ln}\left({n}\right) \\ $$

Question Number 151196 Answers: 0 Comments: 0

Question Number 151164 Answers: 1 Comments: 1

Question Number 151162 Answers: 1 Comments: 2

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