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Question Number 82800    Answers: 4   Comments: 2

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

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

Question Number 82794    Answers: 0   Comments: 0

(d^2 y/dx^2 ) + 5x ((dy/dx))^2 −6y=ln x

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{5}{x}\:\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} −\mathrm{6}{y}={ln}\:{x} \\ $$

Question Number 82792    Answers: 1   Comments: 1

show that if A⊂R^m and B⊂R^n are compact sets. then A×B={(a,b)∈R^(m+n) :a∈A and b∈B}

$${show}\:{that}\:{if}\:{A}\subset\mathbb{R}^{{m}} \:{and}\:{B}\subset\mathbb{R}^{{n}} \:{are}\: \\ $$$${compact}\:{sets}.\: \\ $$$${then}\:{A}×{B}=\left\{\left({a},{b}\right)\in\mathbb{R}^{{m}+{n}} :{a}\in{A}\:{and}\:{b}\in{B}\right\} \\ $$

Question Number 82782    Answers: 1   Comments: 1

(1). 9(9÷3)−6(8÷3)×2 (2). 2(5/3)+3(2/4)

$$\:\: \\ $$$$\:\:\:\left(\mathrm{1}\right).\:\mathrm{9}\left(\mathrm{9}\boldsymbol{\div}\mathrm{3}\right)−\mathrm{6}\left(\mathrm{8}\boldsymbol{\div}\mathrm{3}\right)×\mathrm{2} \\ $$$$\:\:\:\:\left(\mathrm{2}\right).\:\mathrm{2}\frac{\mathrm{5}}{\mathrm{3}}+\mathrm{3}\frac{\mathrm{2}}{\mathrm{4}} \\ $$

Question Number 82816    Answers: 0   Comments: 3

calculate the exact value ∫_0 ^∞ ((cos(x))/(x^2 +1)) dx

$${calculate}\:{the}\:{exact}\:{value}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{cos}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$

Question Number 82815    Answers: 0   Comments: 2

lim_(x→0^+ ) (1/((1+(1/x))^(1/(ln(x))) ))=?

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {{lim}}\:\frac{\mathrm{1}}{\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{\frac{\mathrm{1}}{{ln}\left({x}\right)}} }=? \\ $$

Question Number 82759    Answers: 1   Comments: 0

find the value of (√(2+(√(2+(√(2+(√(2cos 80^o )))))))) =

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2cos}\:\mathrm{80}^{{o}} }}}}\:=\: \\ $$

Question Number 82755    Answers: 1   Comments: 2

calculate ∫_0 ^∞ ((lnx)/((1+x^2 )^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 82753    Answers: 0   Comments: 3

Question Number 82761    Answers: 0   Comments: 7

lim_(x→0) (((cosx)^(1/m) −(cosx)^(1/n) )/x^2 ) [where m and n integer]

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left({cosx}\right)^{\frac{\mathrm{1}}{{m}}} −\left({cosx}\right)^{\frac{\mathrm{1}}{{n}}} }{{x}^{\mathrm{2}} }\:\:\left[{where}\:{m}\:{and}\:{n}\:{integer}\right] \\ $$

Question Number 82768    Answers: 0   Comments: 1

Question Number 82767    Answers: 1   Comments: 1

If ((sin (A+θ))/(sin (B+θ))) = (√((sin 2A)/(sin 2B))) prove that tan^2 θ = tan A.tan B

$$\mathrm{If}\:\frac{\mathrm{sin}\:\left(\mathrm{A}+\theta\right)}{\mathrm{sin}\:\left({B}+\theta\right)}\:=\:\sqrt{\frac{\mathrm{sin}\:\mathrm{2}{A}}{\mathrm{sin}\:\mathrm{2}{B}}} \\ $$$${prove}\:{that}\:\mathrm{tan}\:^{\mathrm{2}} \theta\:=\:\mathrm{tan}\:{A}.\mathrm{tan}\:{B} \\ $$

Question Number 82729    Answers: 1   Comments: 2

Question Number 82726    Answers: 0   Comments: 10

Question Number 82721    Answers: 1   Comments: 2

show that ∫xe^(−x^6 ) sin(x^3 ) dx=((Γ((5/6)))/3) 1F1[(5/6);(3/2);((−1)/4)]

$${show}\:{that}\: \\ $$$$\int{xe}^{−{x}^{\mathrm{6}} } \:{sin}\left({x}^{\mathrm{3}} \right)\:{dx}=\frac{\Gamma\left(\frac{\mathrm{5}}{\mathrm{6}}\right)}{\mathrm{3}}\:\mathrm{1}{F}\mathrm{1}\left[\frac{\mathrm{5}}{\mathrm{6}};\frac{\mathrm{3}}{\mathrm{2}};\frac{−\mathrm{1}}{\mathrm{4}}\right] \\ $$

Question Number 82719    Answers: 2   Comments: 3

If x,y ∈R satisfy in equation x−4(√y) = 2(√(x−y)) . find range of x

$$\mathrm{If}\:\mathrm{x},{y}\:\in\mathbb{R}\:{satisfy}\:{in}\:{equation}\: \\ $$$${x}−\mathrm{4}\sqrt{{y}}\:=\:\mathrm{2}\sqrt{{x}−{y}}\:.\:{find}\:{range}\:{of}\:{x} \\ $$

Question Number 82701    Answers: 1   Comments: 0

2f(x−1) +3f(x+1) ^ = 3x^2 −5x find f(x)

$$\mathrm{2}{f}\left({x}−\mathrm{1}\right)\:+\mathrm{3}{f}\left({x}+\mathrm{1}\right)\overset{\:} {\:}=\:\mathrm{3}{x}^{\mathrm{2}} −\mathrm{5}{x} \\ $$$${find}\:{f}\left({x}\right) \\ $$

Question Number 82700    Answers: 1   Comments: 4

∫ ((2dx)/(3x(√(5x^2 +6)))) ?

$$\int\:\frac{\mathrm{2}{dx}}{\mathrm{3}{x}\sqrt{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{6}}}\:? \\ $$

Question Number 82699    Answers: 0   Comments: 3

lim_(x→1) (((√(ax−a+b))−3)/(x−1)) = −(3/2) find a and b without L′hopital rule

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\sqrt{{ax}−{a}+{b}}−\mathrm{3}}{{x}−\mathrm{1}}\:=\:−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${find}\:{a}\:{and}\:{b}\:{without}\:{L}'{hopital}\:{rule} \\ $$

Question Number 82687    Answers: 0   Comments: 5

Question Number 82682    Answers: 0   Comments: 3

Help me please....!! Lim_(x→0) ((1/(ex)))^(6x) =...

$$\mathrm{Help}\:\mathrm{me}\:\mathrm{please}....!! \\ $$$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{Lim}}\:\left(\frac{\mathrm{1}}{\mathrm{ex}}\right)^{\mathrm{6x}} =... \\ $$

Question Number 82667    Answers: 1   Comments: 0

if a>0 b>0 a≤b show that a^2 ≤(((2ab)/(a+b)))^2 ≤ab≤(((a+b)/2))^2 ≤((a^2 +b^2 )/2)≤b^2

$${if}\:{a}>\mathrm{0}\:\:{b}>\mathrm{0}\:\:{a}\leqslant{b} \\ $$$$ \\ $$$${show}\:{that}\: \\ $$$${a}^{\mathrm{2}} \leqslant\left(\frac{\mathrm{2}{ab}}{{a}+{b}}\right)^{\mathrm{2}} \leqslant{ab}\leqslant\left(\frac{{a}+{b}}{\mathrm{2}}\right)^{\mathrm{2}} \leqslant\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{\mathrm{2}}\leqslant{b}^{\mathrm{2}} \\ $$

Question Number 82665    Answers: 1   Comments: 2

(√(6561(√(6561(√(6561(√(6561(√(...)))))))))) = 3^( 8^x ) (√(6561(√(...)))) (60 time) find x

$$\sqrt{\mathrm{6561}\sqrt{\mathrm{6561}\sqrt{\mathrm{6561}\sqrt{\mathrm{6561}\sqrt{...}}}}}\:=\:\mathrm{3}^{\:\mathrm{8}^{{x}} \:} \\ $$$$\sqrt{\mathrm{6561}\sqrt{...}}\:\:\left(\mathrm{60}\:{time}\right) \\ $$$${find}\:{x} \\ $$

Question Number 82654    Answers: 3   Comments: 1

If sec x + tan x = 2+(√5) find sin x+ cos x ?

$$\boldsymbol{\mathrm{I}}\mathrm{f}\:\mathrm{sec}\:\mathrm{x}\:+\:\mathrm{tan}\:\mathrm{x}\:=\:\mathrm{2}+\sqrt{\mathrm{5}} \\ $$$$\mathrm{find}\:\mathrm{sin}\:\mathrm{x}+\:\mathrm{cos}\:\mathrm{x}\:? \\ $$

Question Number 82647    Answers: 0   Comments: 2

In the rectangular region, −2<x<2, −3<y<3, the surface charge density is given as ρ_s =(x^2 +y^2 +1)^(3/2) . If no other charge is present, find E at P(0,0,1).

$${In}\:{the}\:{rectangular}\:{region},\:−\mathrm{2}<{x}<\mathrm{2}, \\ $$$$−\mathrm{3}<{y}<\mathrm{3},\:{the}\:{surface}\:{charge}\:{density} \\ $$$${is}\:{given}\:{as}\:\rho_{{s}} =\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} .\:{If}\:{no}\:{other} \\ $$$${charge}\:{is}\:{present},\:{find}\:{E}\:{at}\:{P}\left(\mathrm{0},\mathrm{0},\mathrm{1}\right). \\ $$

Question Number 82644    Answers: 1   Comments: 2

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