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Question Number 84460    Answers: 0   Comments: 2

lim_(x→0) ((sin (2+x)−sin (2−x))/x)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\mathrm{2}+\mathrm{x}\right)−\mathrm{sin}\:\left(\mathrm{2}−\mathrm{x}\right)}{\mathrm{x}} \\ $$

Question Number 84459    Answers: 1   Comments: 2

{ ((log_(10) (x)+((log_(10) (x)+8log_(10) (y))/(log_(10) ^2 (x)+log_(10) ^2 (y)))=3)),((log_(10) (y)+((8log_(10) (x)−log_(10) (y))/(log_(10) ^2 (x)+log_(10) ^2 (y)))=0)) :} find x & y

$$\begin{cases}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{x}\right)+\frac{\mathrm{log}_{\mathrm{10}} \left(\mathrm{x}\right)+\mathrm{8log}_{\mathrm{10}} \left(\mathrm{y}\right)}{\mathrm{log}_{\mathrm{10}} ^{\mathrm{2}} \left(\mathrm{x}\right)+\mathrm{log}_{\mathrm{10}} ^{\mathrm{2}} \left(\mathrm{y}\right)}=\mathrm{3}}\\{\mathrm{log}_{\mathrm{10}} \left(\mathrm{y}\right)+\frac{\mathrm{8log}_{\mathrm{10}} \left(\mathrm{x}\right)−\mathrm{log}_{\mathrm{10}} \left(\mathrm{y}\right)}{\mathrm{log}_{\mathrm{10}} ^{\mathrm{2}} \left(\mathrm{x}\right)+\mathrm{log}_{\mathrm{10}} ^{\mathrm{2}} \left(\mathrm{y}\right)}=\mathrm{0}}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{x}\:\&\:\mathrm{y} \\ $$

Question Number 84456    Answers: 0   Comments: 1

y=ln(√((a+sin(x))/(b−sin(x)))) if ((dy/dx))^2 −tan^2 (x)=1 show that a=b

$${y}={ln}\sqrt{\frac{{a}+{sin}\left({x}\right)}{{b}−{sin}\left({x}\right)}} \\ $$$${if}\:\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} −{tan}^{\mathrm{2}} \left({x}\right)=\mathrm{1} \\ $$$${show}\:{that}\:{a}={b} \\ $$

Question Number 84448    Answers: 1   Comments: 0

5^((x^2 −7∣x∣+10)/(x^2 −6x+9)) < 1

$$\mathrm{5}^{\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{7}\mid\mathrm{x}\mid+\mathrm{10}}{\mathrm{x}^{\mathrm{2}} −\mathrm{6x}+\mathrm{9}}} \:<\:\mathrm{1} \\ $$

Question Number 84442    Answers: 1   Comments: 0

Question Number 84441    Answers: 1   Comments: 1

Question Number 84430    Answers: 0   Comments: 4

Question Number 84420    Answers: 1   Comments: 2

Question Number 84415    Answers: 3   Comments: 0

∫ (√(x − (√(4 − x^2 )))) dx

$$\int\:\sqrt{\mathrm{x}\:−\:\sqrt{\mathrm{4}\:−\:\mathrm{x}^{\mathrm{2}} }}\:\:\mathrm{dx} \\ $$

Question Number 84409    Answers: 3   Comments: 4

Question Number 84407    Answers: 1   Comments: 0

dy+2xy dx = xe^(−x^2 ) y^3 dx

$$\mathrm{dy}+\mathrm{2xy}\:\mathrm{dx}\:=\:\mathrm{xe}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{y}^{\mathrm{3}} \:\mathrm{dx} \\ $$$$ \\ $$

Question Number 84404    Answers: 0   Comments: 1

(x^2 −2)(x^2 −4)(x^2 −6)...(x^2 −2020)=1 x=?

$$\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{6}\right)...\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2020}\right)=\mathrm{1} \\ $$$$\mathrm{x}=? \\ $$

Question Number 84399    Answers: 0   Comments: 1

Question Number 84396    Answers: 0   Comments: 2

∫((x(√(x+1)))/(x+2))dx

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

Question Number 84395    Answers: 0   Comments: 0

((x(√(x+1)))/(x+2))

$$\frac{\mathrm{x}\sqrt{\mathrm{x}+\mathrm{1}}}{\mathrm{x}+\mathrm{2}} \\ $$

Question Number 84394    Answers: 0   Comments: 2

find the solution ((2x)/(x−2)) ≤ ∣x−3∣

$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution} \\ $$$$\frac{\mathrm{2x}}{\mathrm{x}−\mathrm{2}}\:\leqslant\:\mid\mathrm{x}−\mathrm{3}\mid\: \\ $$

Question Number 84393    Answers: 0   Comments: 0

if x^x .y^y .z^z =x^y .y^z .z^x =x^z .y^x .z^y such that x, y and z are positive intigers greater than 1 ,what is the value of xyz and x+y+z ?

$${if}\:{x}^{{x}} .{y}^{{y}} .{z}^{{z}} ={x}^{{y}} .{y}^{{z}} .{z}^{{x}} ={x}^{{z}} .{y}^{{x}} .{z}^{{y}} \:{such}\:{that}\:{x},\:{y}\:{and}\:{z}\: \\ $$$${are}\:{positive}\:{intigers}\:{greater}\:{than}\:\mathrm{1} \\ $$$$,{what}\:{is}\:{the}\:{value}\:{of}\:{xyz}\:{and}\:{x}+{y}+{z}\:? \\ $$

Question Number 84386    Answers: 1   Comments: 0

∫(√(x−(√(4−x^2 )))) dx

$$\int\sqrt{{x}−\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 84384    Answers: 0   Comments: 3

[x]^x =2(√2) , ∀x>0

$$\left[{x}\right]^{{x}} =\mathrm{2}\sqrt{\mathrm{2}}\:\:,\:\forall{x}>\mathrm{0} \\ $$

Question Number 84382    Answers: 0   Comments: 0

∫((cos(2x) sin(x))/(cos(x)+sin(2x))) dx

$$\int\frac{{cos}\left(\mathrm{2}{x}\right)\:{sin}\left({x}\right)}{{cos}\left({x}\right)+{sin}\left(\mathrm{2}{x}\right)}\:{dx} \\ $$

Question Number 84381    Answers: 1   Comments: 3

Question Number 84379    Answers: 2   Comments: 0

Question Number 84377    Answers: 1   Comments: 0

lim_(x→0) (((√(1+tan x))−(√(1+sin x)))/(x^2 sin x))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{tan}\:\mathrm{x}}−\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\mathrm{2}} \mathrm{sin}\:\mathrm{x}} \\ $$

Question Number 84370    Answers: 2   Comments: 0

1.) ∣x∣ +∣x+2∣ <5 2.) ∣x∣ +∣x+2∣ + ∣2−x∣ ≤8

$$\left.\mathrm{1}.\right)\:\mid{x}\mid\:+\mid{x}+\mathrm{2}\mid\:<\mathrm{5} \\ $$$$\left.\mathrm{2}.\right)\:\mid{x}\mid\:+\mid{x}+\mathrm{2}\mid\:+\:\mid\mathrm{2}−{x}\mid\:\leqslant\mathrm{8} \\ $$

Question Number 84446    Answers: 0   Comments: 0

Question Number 84367    Answers: 0   Comments: 0

Given the squence of R (x_n ). If Σ_(n=1) ^∞ x_k <∞, Find lim_(n→∞) ((Σ_(k=1) ^∞ (√x_k ))/(√n))=.....

$$ \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Given}\:\mathrm{the}\:\mathrm{squence}\:\mathrm{of}\:\mathbb{R} \\ $$$$\left({x}_{{n}} \right).\:{If}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{x}_{{k}} <\infty, \\ $$$${Find} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\sqrt{{x}_{{k}} }}{\sqrt{{n}}}=..... \\ $$$$ \\ $$

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