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Question Number 82867 Answers: 0 Comments: 3

hello prove that ∫_0 ^(+∞) sin(x^4 )dx=sin((π/8))∫_0 ^(+∞) e^(−x^4 ) dx? verry nice day Good Bless You

$${hello}\:{prove}\:{that}\:\int_{\mathrm{0}} ^{+\infty} {sin}\left({x}^{\mathrm{4}} \right){dx}={sin}\left(\frac{\pi}{\mathrm{8}}\right)\int_{\mathrm{0}} ^{+\infty} {e}^{−{x}^{\mathrm{4}} } {dx}? \\ $$$${verry}\:{nice}\:{day}\:{Good}\:{Bless}\:{You} \\ $$

Question Number 82843 Answers: 1 Comments: 0

show that (((1+(√3) i)^4 (1+i)^8 )/((cos100°−i sin100)^3 ))=−256

$${show}\:{that} \\ $$$$\frac{\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{i}\right)^{\mathrm{4}} \left(\mathrm{1}+{i}\right)^{\mathrm{8}} }{\left({cos}\mathrm{100}°−{i}\:{sin}\mathrm{100}\right)^{\mathrm{3}} }=−\mathrm{256} \\ $$

Question Number 82839 Answers: 0 Comments: 8

if the first and fifth terms of arithmetic peogression are equal and the seventh and fourtenth terms of another arithmetic are equal then show that the first term from the first arithmetic is equal the tenth from the second one and so sorry because my english is not so good

$${if}\:{the}\:{first}\:{and}\:{fifth}\:{terms}\:{of}\:{arithmetic} \\ $$$${peogression}\:{are}\:{equal}\:{and}\:{the}\:{seventh} \\ $$$${and}\:{fourtenth}\:{terms}\:{of}\:{another}\:{arithmetic}\:{are} \\ $$$${equal}\:{then}\:{show}\:{that}\:{the}\:{first}\:{term}\:{from} \\ $$$${the}\:{first}\:{arithmetic}\:{is}\:{equal}\:{the}\:{tenth} \\ $$$${from}\:{the}\:{second}\:{one} \\ $$$${and}\:{so}\:{sorry}\:{because}\:{my}\:{english}\:{is} \\ $$$${not}\:{so}\:{good} \\ $$

Question Number 82877 Answers: 1 Comments: 1

1)find xy∈R 2)find x,y∈Z (x+2yi)^6 =8i

$$\left.\mathrm{1}\right){find}\:{xy}\in{R} \\ $$$$\left.\mathrm{2}\right){find}\:{x},{y}\in{Z} \\ $$$$\left({x}+\mathrm{2}{yi}\right)^{\mathrm{6}} =\mathrm{8}{i} \\ $$

Question Number 82821 Answers: 0 Comments: 1

Log_y x+Log_x y =64 find x and y

$${Log}_{{y}} \:{x}+{Log}_{{x}} \:{y}\:=\mathrm{64} \\ $$$${find}\:{x}\:{and}\:{y} \\ $$

Question Number 82806 Answers: 1 Comments: 1

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

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