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Question Number 145073 Answers: 1 Comments: 2

let a_1 ,a_2 ,...,a_n be positive real numbers such that a_1 +a_2 +...+a_n =1 then find maximum value of a_1 ^a_1 .a_2 ^a_2 ....a_n ^a_n ?

Question Number 145071 Answers: 1 Comments: 0

log _(((x/2))) (x+2) = 1+ log _x (4−x) x=?

$$\mathrm{log}\:_{\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} \left(\mathrm{x}+\mathrm{2}\right)\:=\:\mathrm{1}+\:\mathrm{log}\:_{\mathrm{x}} \left(\mathrm{4}−\mathrm{x}\right) \\ $$$$\:\mathrm{x}=? \\ $$

Question Number 145067 Answers: 1 Comments: 0

log _3 (x+1) =log _4 (x+8) x=?

$$\:\mathrm{log}\:_{\mathrm{3}} \left({x}+\mathrm{1}\right)\:=\mathrm{log}\:_{\mathrm{4}} \left({x}+\mathrm{8}\right) \\ $$$$\:{x}=? \\ $$

Question Number 145065 Answers: 0 Comments: 0

(1/( (√(1+x))))=1+Σ_(k=1) ^n [(((−1)^k )/2^(2k) )C_(2k) ^k ]x^k +o(x^n )

$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}}}=\mathrm{1}+\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left[\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{2}^{\mathrm{2k}} }\mathrm{C}_{\mathrm{2k}} ^{\mathrm{k}} \right]\mathrm{x}^{\mathrm{k}} +\mathrm{o}\left(\mathrm{x}^{\mathrm{n}} \right) \\ $$

Question Number 145064 Answers: 1 Comments: 0

∫cos 2xln (1+tan x)dx

$$\:\:\:\:\:\:\int\mathrm{cos}\:\mathrm{2xln}\:\left(\mathrm{1}+\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$$$\:\:\:\:\:\: \\ $$

Question Number 145057 Answers: 0 Comments: 3

find k if y^2 +y+k is a perfect square

$$\boldsymbol{{find}}\:\:\boldsymbol{{k}}\:\:\boldsymbol{{if}}\:\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{y}}+\boldsymbol{{k}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{perfect}} \\ $$$$\boldsymbol{{square}} \\ $$

Question Number 145052 Answers: 0 Comments: 2

find laurant series at f(z)=(1/(lnz))????

$${find}\:{laurant}\:{series}\:{at}\:{f}\left({z}\right)=\frac{\mathrm{1}}{{lnz}}???? \\ $$

Question Number 145047 Answers: 3 Comments: 1

Question Number 145046 Answers: 0 Comments: 0

find ∫_0 ^∞ ∣cosx +sinx∣e^(−x) dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\mid\mathrm{cosx}\:+\mathrm{sinx}\mid\mathrm{e}^{−\mathrm{x}} \:\mathrm{dx} \\ $$

Question Number 145044 Answers: 3 Comments: 0

calculate ∫_0 ^∞ e^(−2x) (√(1+sinx))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\mathrm{e}^{−\mathrm{2x}} \sqrt{\mathrm{1}+\mathrm{sinx}}\mathrm{dx} \\ $$

Question Number 145025 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (1/((n+1)(√n)+n(√(n+1)))) = ?

$$\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\sqrt{{n}}+{n}\sqrt{{n}+\mathrm{1}}}\:=\:? \\ $$

Question Number 145023 Answers: 0 Comments: 4

Question Number 145021 Answers: 2 Comments: 0

Question Number 145019 Answers: 1 Comments: 0

Solve the equation: y+1=x^6 −9x^4 +6x^2 ; x;y∈P

$${Solve}\:{the}\:{equation}: \\ $$$${y}+\mathrm{1}={x}^{\mathrm{6}} −\mathrm{9}{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{2}} \:\:;\:\:{x};{y}\in{P} \\ $$

Question Number 145012 Answers: 1 Comments: 0

Question Number 145011 Answers: 2 Comments: 0

Question Number 145010 Answers: 2 Comments: 0

Question Number 145009 Answers: 3 Comments: 0

Question Number 145004 Answers: 0 Comments: 1

Question Number 145003 Answers: 0 Comments: 0

Solve the equation: 2^a 3^b 7^c = c^2 cbb^6 ^(−) ; a;b;c∈P

$${Solve}\:{the}\:{equation}: \\ $$$$\mathrm{2}^{\boldsymbol{{a}}} \mathrm{3}^{\boldsymbol{{b}}} \mathrm{7}^{\boldsymbol{{c}}} \:=\:\overline {\boldsymbol{{c}}^{\mathrm{2}} \boldsymbol{{cbb}}^{\mathrm{6}} \:}\:\:;\:\:\boldsymbol{{a}};\boldsymbol{{b}};\boldsymbol{{c}}\in\boldsymbol{{P}} \\ $$

Question Number 145002 Answers: 0 Comments: 2

Find the last digit of the number: 1^(1989) +2^(1989) +3^(1989) +...+1989^(1989)

$${Find}\:{the}\:{last}\:{digit}\:{of}\:{the}\:{number}: \\ $$$$\mathrm{1}^{\mathrm{1989}} +\mathrm{2}^{\mathrm{1989}} +\mathrm{3}^{\mathrm{1989}} +...+\mathrm{1989}^{\mathrm{1989}} \\ $$

Question Number 145000 Answers: 0 Comments: 1

If F(x+1)=F(x−1)=x^2 then F^(−1) (x)=?

$$\mathrm{If}\:\mathrm{F}\left(\mathrm{x}+\mathrm{1}\right)=\mathrm{F}\left(\mathrm{x}−\mathrm{1}\right)=\mathrm{x}^{\mathrm{2}} \\ $$$$\mathrm{then}\:\mathrm{F}^{−\mathrm{1}} \left(\mathrm{x}\right)=? \\ $$

Question Number 144998 Answers: 1 Comments: 0

cos^2 1°+cos^2 2°+cos^2 3°+...+cos^2 360° = ?

$$\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{1}°+\mathrm{cos}\:^{\mathrm{2}} \mathrm{2}°+\mathrm{cos}\:^{\mathrm{2}} \mathrm{3}°+...+\mathrm{cos}\:^{\mathrm{2}} \mathrm{360}°\:=\:? \\ $$

Question Number 144997 Answers: 0 Comments: 0

∫_0 ^1 ((ln(1+x)ln(1+x^2 ))/x)dx=(π/2)G−((33)/(32))ζ(3)

$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}}\mathrm{dx}=\frac{\pi}{\mathrm{2}}\mathrm{G}−\frac{\mathrm{33}}{\mathrm{32}}\zeta\left(\mathrm{3}\right) \\ $$

Question Number 144996 Answers: 0 Comments: 0

Question Number 144995 Answers: 1 Comments: 0

Find the maximum distance between two points on the curve (x^4 /a^4 ) + (y^4 /b^4 ) = 1 .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{distance}\: \\ $$$$\mathrm{between}\:\mathrm{two}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\: \\ $$$$\:\mathrm{curve}\:\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{a}^{\mathrm{4}} }\:+\:\frac{\mathrm{y}^{\mathrm{4}} }{\mathrm{b}^{\mathrm{4}} }\:=\:\mathrm{1}\:. \\ $$

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