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Question Number 63065    Answers: 0   Comments: 3

If I=∫_( 0) ^1 (dx/(√(1+x^4 ))) , then

$$\mathrm{If}\:{I}=\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}\:,\:\mathrm{then} \\ $$

Question Number 63076    Answers: 0   Comments: 0

show that f:A→B is bijection then f(A_1 ^c )=[f(A_1 )]^c

$${show}\:{that}\:{f}:{A}\rightarrow{B}\:{is}\:{bijection}\:{then}\:{f}\left({A}_{\mathrm{1}} ^{{c}} \right)=\left[{f}\left({A}_{\mathrm{1}} \right)\right]^{{c}} \\ $$

Question Number 63059    Answers: 0   Comments: 0

Question Number 63054    Answers: 0   Comments: 0

if Σ∣a_n ∣ is convergent, then prove that there exists a subsequence {n_k a_n_k } with lim_(k→∞) n_k a_n_k =0

$${if}\:\Sigma\mid{a}_{{n}} \:\mid\:{is}\:{convergent},\:{then} \\ $$$${prove}\:{that}\:{there}\:{exists}\: \\ $$$${a}\:{subsequence}\:\left\{{n}_{{k}} {a}_{{n}_{{k}} } \right\}\:\:{with} \\ $$$$\underset{{k}\rightarrow\infty} {\mathrm{lim}}{n}_{{k}} {a}_{{n}_{{k}} } =\mathrm{0} \\ $$

Question Number 63034    Answers: 0   Comments: 0

calculate ∫_0 ^(π/2) (ln(cosx))^2 dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({ln}\left({cosx}\right)\right)^{\mathrm{2}} \:{dx}\: \\ $$

Question Number 63033    Answers: 0   Comments: 1

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

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

Question Number 63032    Answers: 0   Comments: 1

let f(z) =(1/(sin(πz))) calculate Res(f,n) with n integr

$${let}\:{f}\left({z}\right)\:=\frac{\mathrm{1}}{{sin}\left(\pi{z}\right)}\:\:{calculate}\:{Res}\left({f},{n}\right)\:{with}\:{n}\:{integr} \\ $$

Question Number 63031    Answers: 0   Comments: 2

let f(z) =((sin(z))/z^2 ) calculate Res(f,0)

$${let}\:{f}\left({z}\right)\:=\frac{{sin}\left({z}\right)}{{z}^{\mathrm{2}} }\:\:{calculate}\:{Res}\left({f},\mathrm{0}\right) \\ $$

Question Number 63026    Answers: 0   Comments: 0

calculate ∫_0 ^π ((sin(2x))/(2cosx −3sinx))dx

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}{cosx}\:−\mathrm{3}{sinx}}{dx} \\ $$

Question Number 63023    Answers: 0   Comments: 1

calculate ∫_(−∞) ^(+∞) ((x^2 −3)/(x^4 +x^2 +1))dx .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} −\mathrm{3}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}{dx}\:. \\ $$

Question Number 63021    Answers: 2   Comments: 3

solve this equation x^y =y^x x,y∈R.

$${solve}\:{this}\:{equation} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}^{{y}} ={y}^{{x}} \\ $$$$ \\ $$$$ \\ $$$${x},{y}\in\mathbb{R}. \\ $$

Question Number 63017    Answers: 0   Comments: 0

445x((5x)/)

$$\mathrm{445}\boldsymbol{{x}}\frac{\mathrm{5}\boldsymbol{{x}}}{} \\ $$$$ \\ $$

Question Number 63016    Answers: 1   Comments: 0

The sides of a hexagon are enlarged by three times. Find the ratio of the areas of the new and old hexagon

$$\mathrm{The}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hexagon}\:\mathrm{are}\:\mathrm{enlarged}\:\mathrm{by}\: \\ $$$$\mathrm{three}\:\mathrm{times}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{areas} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{new}\:\mathrm{and}\:\mathrm{old}\:\mathrm{hexagon} \\ $$

Question Number 63015    Answers: 0   Comments: 0

f(x,y,z)= x(p+z)+y(p−z) +((4x^3 )/(p+z))+((4y^3 )/(p−z))+4(x+y)^2 (y−x) ∀ p(x,y)=c+(x−y)(√(1+(x+y)^2 )) +(x^2 −y^2 ) Determine x,y,z such that f is maximum. (c is a constant). Assume y≥x.

$${f}\left({x},{y},{z}\right)=\:{x}\left({p}+{z}\right)+{y}\left({p}−{z}\right) \\ $$$$\:\:\:\:+\frac{\mathrm{4}{x}^{\mathrm{3}} }{{p}+{z}}+\frac{\mathrm{4}{y}^{\mathrm{3}} }{{p}−{z}}+\mathrm{4}\left({x}+{y}\right)^{\mathrm{2}} \left({y}−{x}\right) \\ $$$$\forall\:\:{p}\left({x},{y}\right)={c}+\left({x}−{y}\right)\sqrt{\mathrm{1}+\left({x}+{y}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right) \\ $$$${Determine}\:{x},{y},{z}\:{such}\:{that}\:{f}\:{is} \\ $$$${maximum}.\:\left({c}\:{is}\:{a}\:{constant}\right). \\ $$$${Assume}\:{y}\geqslant{x}. \\ $$

Question Number 62998    Answers: 0   Comments: 12

Solve for x: 5^x +6x=7

$${Solve}\:{for}\:{x}:\:\:\mathrm{5}^{\boldsymbol{{x}}} +\mathrm{6}\boldsymbol{{x}}=\mathrm{7} \\ $$

Question Number 62997    Answers: 0   Comments: 1

∫((ln(1+xsin^2 (x)))/(sin^2 (x))) dx

$$\int\frac{{ln}\left(\mathrm{1}+{xsin}^{\mathrm{2}} \left({x}\right)\right)}{{sin}^{\mathrm{2}} \left({x}\right)}\:{dx} \\ $$

Question Number 62995    Answers: 0   Comments: 0

Question Number 62987    Answers: 0   Comments: 0

Question Number 62983    Answers: 3   Comments: 0

If tan 2θ tan θ = 1, then θ =

$$\mathrm{If}\:\:\:\mathrm{tan}\:\mathrm{2}\theta\:\mathrm{tan}\:\theta\:=\:\mathrm{1},\:\mathrm{then}\:\theta\:= \\ $$

Question Number 62982    Answers: 1   Comments: 0

((tg((x/2))−1)/(ctg((x/2))))=2sin((x/2)) x∈(180°;540°)

$$\frac{\boldsymbol{\mathrm{tg}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)−\mathrm{1}}{\boldsymbol{\mathrm{ctg}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)}=\mathrm{2}\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)\:\:\boldsymbol{\mathrm{x}}\in\left(\mathrm{180}°;\mathrm{540}°\right) \\ $$$$ \\ $$

Question Number 62981    Answers: 1   Comments: 1

Question Number 62970    Answers: 1   Comments: 1

Question Number 62945    Answers: 1   Comments: 0

Find the greatest coefficient in the expansion of (6 − 4x)^(−3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\:\left(\mathrm{6}\:−\:\mathrm{4x}\right)^{−\mathrm{3}} \\ $$

Question Number 62942    Answers: 1   Comments: 10

Make r the subject of the formular: S = ((a(r^n − 1))/(r − 1))

$$\mathrm{Make}\:\:\mathrm{r}\:\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\mathrm{the}\:\mathrm{formular}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{S}\:\:=\:\:\frac{\mathrm{a}\left(\mathrm{r}^{\mathrm{n}} \:−\:\mathrm{1}\right)}{\mathrm{r}\:−\:\mathrm{1}} \\ $$

Question Number 62938    Answers: 1   Comments: 1

Question Number 62937    Answers: 1   Comments: 3

∫_0 ^( x) (1/(1+x^2 )) dx

$$\int_{\mathrm{0}} ^{\:\:\mathrm{x}} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

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