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Question Number 82514 Answers: 0 Comments: 1

Find maximum of n such that n^2 + n + 1 is factor of 19n^(2019) + 1 .

$${Find}\:\:\:{maximum}\:\:{of}\:\:{n}\:\:{such}\:\:{that}\:\: \\ $$$${n}^{\mathrm{2}} \:+\:{n}\:+\:\mathrm{1}\:\:{is}\:\:{factor}\:\:{of}\:\:\mathrm{19}{n}^{\mathrm{2019}} \:+\:\mathrm{1}\:. \\ $$

Question Number 82510 Answers: 1 Comments: 1

∫ (√(x+(√x) )) dx = ?

$$\int\:\sqrt{{x}+\sqrt{{x}}\:}\:{dx}\:=\:? \\ $$

Question Number 82499 Answers: 1 Comments: 3

Question Number 82498 Answers: 0 Comments: 0

Question Number 82497 Answers: 0 Comments: 0

Question Number 82494 Answers: 0 Comments: 4

Find the limit _(x→0) ^(lim) (((tan x)/x))^(1/x^( 2) )

$$\:\: \\ $$$$\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{limit}} \\ $$$$\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\overset{\boldsymbol{\mathrm{lim}}} {\:}}\:\left(\frac{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}\right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\:\mathrm{2}} }} \\ $$

Question Number 82489 Answers: 1 Comments: 0

If a^b =b^a and a=2b then find the value of a^2 +b^2 .

$$ \\ $$$$\: \\ $$$$\mathrm{If}\:\mathrm{a}^{\mathrm{b}} =\mathrm{b}^{\mathrm{a}} \:\mathrm{and}\:\mathrm{a}=\mathrm{2b}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\ $$$$\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \:. \\ $$

Question Number 82485 Answers: 2 Comments: 4

Question Number 82476 Answers: 1 Comments: 0

solve xy^(′′) +2y^′ +xy=0 with initial conditions y(o)=1 and y^′ (1)=0

$${solve}\:\:{xy}^{''} \:+\mathrm{2}{y}^{'} \:+{xy}=\mathrm{0}\:{with}\:{initial}\:{conditions} \\ $$$${y}\left({o}\right)=\mathrm{1}\:{and}\:{y}^{'} \left(\mathrm{1}\right)=\mathrm{0} \\ $$

Question Number 82474 Answers: 0 Comments: 2

∫_(−1/2) ^(1/2) ∣ x cos(((πx)/2))∣ dx =

$$\underset{−\mathrm{1}/\mathrm{2}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\mid\:{x}\:\mathrm{cos}\left(\frac{\pi{x}}{\mathrm{2}}\right)\mid\:{dx}\:= \\ $$

Question Number 82659 Answers: 1 Comments: 3

Question Number 82456 Answers: 0 Comments: 6

Question Number 82452 Answers: 0 Comments: 1

nature ofthe serie Σ_(n=0) ^∞ ln(cos((1/2^n )))

$${nature}\:{ofthe}\:{serie}\:\sum_{{n}=\mathrm{0}} ^{\infty} {ln}\left({cos}\left(\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\right)\right) \\ $$

Question Number 82450 Answers: 1 Comments: 1

∫_0 ^π (1/(1+(tan(x))^(√2) )) dx

$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{1}}{\mathrm{1}+\left({tan}\left({x}\right)\right)^{\sqrt{\mathrm{2}}} }\:{dx} \\ $$

Question Number 82448 Answers: 0 Comments: 2

Lim_(x→(π/2)) {((sin (6x−3π)^2 −sin (6x−3π)sin (4x−2π))/(5x^2 cos (5x−((5π)/2) )))

$$\underset{\mathrm{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{Lim}}\:\left\{\frac{\mathrm{sin}\:\left(\mathrm{6x}−\mathrm{3}\pi\right)^{\mathrm{2}} −\mathrm{sin}\:\left(\mathrm{6x}−\mathrm{3}\pi\right)\mathrm{sin}\:\left(\mathrm{4x}−\mathrm{2}\pi\right)}{\mathrm{5x}^{\mathrm{2}} \:\mathrm{cos}\:\left(\mathrm{5x}−\frac{\mathrm{5}\pi}{\mathrm{2}}\:\right)}\right. \\ $$

Question Number 82447 Answers: 1 Comments: 0

calculate Σ_(n=2) ^∞ (((−1)^n )/n)ξ(n)

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\xi\left({n}\right) \\ $$

Question Number 82446 Answers: 0 Comments: 3

show that ∫_0 ^∞ x^(−log(x)) x log(x) dx=e(√π)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{−{log}\left({x}\right)} \:{x}\:{log}\left({x}\right)\:{dx}={e}\sqrt{\pi} \\ $$

Question Number 82445 Answers: 1 Comments: 0

calculate Σ_(n=2) ^∞ ((ξ(n)−1)/n) with ξ(x) =Σ_(n=1) ^∞ (1/n^x ) (x>1)

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\xi\left({n}\right)−\mathrm{1}}{{n}} \\ $$$${with}\:\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\left({x}>\mathrm{1}\right) \\ $$

Question Number 82440 Answers: 0 Comments: 1

find ∫ ((x+1)/(x+2))(√((1−x)/(1+x)))dx

$${find}\:\int\:\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx} \\ $$

Question Number 82439 Answers: 0 Comments: 1

calculate I_n =∫_0 ^1 x^n (√(1+x+x^2 ))dx

$${calculate}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 82442 Answers: 0 Comments: 1

1)find ∫ ((√(x^2 −x+1))/(x^2 +3))dx 2)calculate ∫_0 ^1 ((√(x^2 −x+1))/(x^2 +3))dx

$$\left.\mathrm{1}\right){find}\:\int\:\frac{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

Question Number 82435 Answers: 0 Comments: 1

calculate ∫_4 ^(+∞) (x^3 /((2x+1)^3 (x−3)^5 ))dx

$${calculate}\:\int_{\mathrm{4}} ^{+\infty} \:\:\:\:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} }{dx} \\ $$

Question Number 82434 Answers: 0 Comments: 0

1)decompose inside C(x)and R(x) the fraction F(x)=((2x+1)/((x^2 +1)^3 (x−1)^2 )) 2) find the value of ∫_3 ^(+∞) F(x)dx

$$\left.\mathrm{1}\right){decompose}\:{inside}\:{C}\left({x}\right){and}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\frac{\mathrm{2}{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} \left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{3}} ^{+\infty} {F}\left({x}\right){dx} \\ $$

Question Number 82433 Answers: 0 Comments: 1

1)decompose inside C(x)and R(x) F=(1/((x^2 +x+1)^2 )) 2)calculate ∫_0 ^∞ (dx/((x^2 +x+1)^2 ))

$$\left.\mathrm{1}\right){decompose}\:{inside}\:{C}\left({x}\right){and}\:{R}\left({x}\right)\:{F}=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 82431 Answers: 1 Comments: 5

Question Number 82426 Answers: 0 Comments: 2

Lim ((e^x −1−x^2 )/(x^4 +x^3 +x^2 )) = ... x→0

$$\mathrm{Lim}\:\frac{\mathrm{e}^{\mathrm{x}} −\mathrm{1}−\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} }\:=\:... \\ $$$$\mathrm{x}\rightarrow\mathrm{0} \\ $$

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