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AllQuestion and Answers: Page 1611

Question Number 48090 Answers: 1 Comments: 0

Question Number 48078 Answers: 1 Comments: 0

Question Number 48075 Answers: 1 Comments: 1

solve (∣x^2 −1∣−(1/2))x+((√6)/(18))=0

$$\mathrm{solve}\:\:\:\:\:\left(\mid{x}^{\mathrm{2}} −\mathrm{1}\mid−\frac{\mathrm{1}}{\mathrm{2}}\right){x}+\frac{\sqrt{\mathrm{6}}}{\mathrm{18}}=\mathrm{0} \\ $$

Question Number 48068 Answers: 1 Comments: 1

let u_n =∫_0 ^∞ (dt/(1+t^n )) find nature of Σ u_n and Σ (u_n /n^2 ) and Σ (u_n /n^3 )

$${let}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\mathrm{1}+{t}^{{n}} } \\ $$$${find}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:\:\:\:{and}\:\Sigma\:\frac{{u}_{{n}} }{{n}^{\mathrm{2}} }\:\:{and}\:\Sigma\:\frac{{u}_{{n}} }{{n}^{\mathrm{3}} } \\ $$

Question Number 48067 Answers: 0 Comments: 1

let y>0 give ∫_0 ^∞ (x^y /(e^x −1))dx at form of series.

$${let}\:{y}>\mathrm{0}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{y}} }{{e}^{{x}} −\mathrm{1}}{dx}\:{at}\:{form}\:{of}\:{series}. \\ $$

Question Number 48066 Answers: 1 Comments: 4

(√(1/2)).(√((1/2)+(1/2)(√(1/2)))).(√((1/2)+(1/2)(√((1/2)+(1/2)(√(1/2))))))......∞=?

$$\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}.\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}.\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}}......\infty=? \\ $$

Question Number 48065 Answers: 0 Comments: 1

let f : ]0,1[ contnue integrable u_n =(−1)^n ∫_0 ^1 x^n f(x)dx prove that Σ u_n cnverge and find its sum

$$\left.{let}\:{f}\:\:\:\::\:\:\right]\mathrm{0},\mathrm{1}\left[\:\:{contnue}\:{integrable}\:\:{u}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} {f}\left({x}\right){dx}\right. \\ $$$${prove}\:{that}\:\Sigma\:{u}_{{n}} \:{cnverge}\:{and}\:{find}\:{its}\:{sum} \\ $$$$ \\ $$

Question Number 48064 Answers: 1 Comments: 1

calculate A =∫_0 ^1 (1+x^2 )(√(1−x^2 ))dx −∫_0 ^1 (1−x^2 )(√(1+x^2 ))dx

$${calculate}\:{A}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx}\:\:−\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 48063 Answers: 0 Comments: 0

let W(x) =∫_(−∞) ^(+∞) ((arctan(xt^2 ))/(2+t^2 ))dt 1) find a explicit form of f(x) 2) find the value of ∫_(−∞) ^(+∞) (t^2 /((2+t^2 )(1+x^2 t^4 )))dt .

$${let}\:{W}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{t}^{\mathrm{2}} }{\left(\mathrm{2}+{t}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{4}} \right)}{dt}\:. \\ $$

Question Number 48062 Answers: 0 Comments: 0

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

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

Question Number 48057 Answers: 2 Comments: 1

Question Number 48052 Answers: 2 Comments: 1

Question Number 48050 Answers: 1 Comments: 1

Question Number 48055 Answers: 2 Comments: 6

Solve the system: { ((x^3 +x^2 y−4xy^2 −4y^3 =0)),((x^2 −2xy−3y^2 −x−y=0)) :}

$${Solve}\:{the}\:{system}: \\ $$$$\begin{cases}{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} {y}−\mathrm{4}{xy}^{\mathrm{2}} −\mathrm{4}{y}^{\mathrm{3}} =\mathrm{0}}\\{{x}^{\mathrm{2}} −\mathrm{2}{xy}−\mathrm{3}{y}^{\mathrm{2}} −{x}−{y}=\mathrm{0}}\end{cases} \\ $$

Question Number 48043 Answers: 0 Comments: 1

let f(α) =∫_(−∞) ^(+∞) ((cos(αx^3 ))/(x^2 +8)) dx 1)calculate f(α) 2) calculate ∫_(−∞) ^(+∞) ((cos(2x^3 ))/(x^2 +8))dx .

$${let}\:{f}\left(\alpha\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left(\alpha{x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{8}}\:{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}\left(\alpha\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left(\mathrm{2}{x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{8}}{dx}\:. \\ $$

Question Number 48042 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((2x+1)/((x^2 +i)(x^2 +4)))dx (i^2 =−1)

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{2}{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} +{i}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)}{dx}\:\:\:\left({i}^{\mathrm{2}} =−\mathrm{1}\right) \\ $$

Question Number 48040 Answers: 0 Comments: 1

let f(α)=∫_(−∞) ^(+∞) ((xsin(αx))/((1+x^2 )^2 ))dx calculate f(α) and f^′ (α).(α from R) .

$${let}\:{f}\left(\alpha\right)=\int_{−\infty} ^{+\infty} \:\:\frac{{xsin}\left(\alpha{x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$${calculate}\:{f}\left(\alpha\right)\:{and}\:{f}^{'} \left(\alpha\right).\left(\alpha\:{from}\:{R}\right)\:. \\ $$

Question Number 48027 Answers: 2 Comments: 1

Question Number 48009 Answers: 0 Comments: 0

let f_n (t)=t^(n−1) sin(nθ) with t from[0,1[ and θ from [0,π[ 1) prove the uniform convergence of Σ f_n (t) on [0,1[ 2) let S(t)=Σ f_n (t) calculate ∫_0 ^1 S(t)dt.

$${let}\:\:\:{f}_{{n}} \left({t}\right)={t}^{{n}−\mathrm{1}} {sin}\left({n}\theta\right)\:{with}\:{t}\:{from}\left[\mathrm{0},\mathrm{1}\left[\:{and}\:\:\theta\:{from}\:\left[\mathrm{0},\pi\left[\right.\right.\right.\right. \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{uniform}\:{convergence}\:{of}\:\Sigma\:{f}_{{n}} \left({t}\right)\:{on}\:\left[\mathrm{0},\mathrm{1}\left[\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{S}\left({t}\right)=\Sigma\:{f}_{{n}} \left({t}\right)\:\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {S}\left({t}\right){dt}. \\ $$

Question Number 47993 Answers: 0 Comments: 5

Solve in R ((59+(√(x−2))))^(1/3) +((12−(√(x−11))))^(1/3) =6

$${Solve}\:{in}\:\mathbb{R} \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{59}+\sqrt{{x}−\mathrm{2}}}+\sqrt[{\mathrm{3}}]{\mathrm{12}−\sqrt{{x}−\mathrm{11}}}=\mathrm{6} \\ $$

Question Number 47986 Answers: 3 Comments: 1

Question Number 47985 Answers: 1 Comments: 3

calculate I=∫_0 ^1 (√(1+2(√(x−x^2 ))))dx and J =∫_0 ^1 (√(1−2(√(x−x^2 ))))dx

$${calculate}\:{I}=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{2}\sqrt{{x}−{x}^{\mathrm{2}} }}{dx}\:\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−\mathrm{2}\sqrt{{x}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 47976 Answers: 0 Comments: 1

Question Number 47967 Answers: 1 Comments: 0

A particle moves in a linear scare such that acceleration after t seconds is a ms^(−2) where a= 2t^2 + t.If its initial velocity was 3ms^(−1) find an expression for S,the distance in meters traveled from start t seconds.

$${A}\:{particle}\:{moves}\:{in}\:{a}\:{linear}\:{scare}\:{such}\:{that}\:{acceleration} \\ $$$${after}\:{t}\:{seconds}\:{is}\:{a}\:{ms}^{−\mathrm{2}} \:{where}\:{a}=\:\mathrm{2}{t}^{\mathrm{2}} +\:{t}.{If}\:{its}\:{initial}\: \\ $$$${velocity}\:{was}\:\mathrm{3}{ms}^{−\mathrm{1}} \:{find}\:{an}\:{expression}\:{for}\:{S},{the}\:{distance}\:{in}\:{meters} \\ $$$${traveled}\:{from}\:{start}\:{t}\:{seconds}. \\ $$

Question Number 47966 Answers: 1 Comments: 1

a(x^2 +y^2 )+b(x+y)= c & x^2 −y^2 = R^2 Solve for x or y .

$${a}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+{b}\left({x}+{y}\right)=\:{c} \\ $$$$\:\&\:\:\:\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:=\:{R}^{\mathrm{2}} \\ $$$${Solve}\:{for}\:{x}\:{or}\:{y}\:. \\ $$

Question Number 47965 Answers: 0 Comments: 0

A small body is made to travel linearly t sconds after the start.Its distance S meters from a fix point O on a linear scale is given by S = t^2 −5t + 6. a) How far is the body from O at the start? b)with what velocity does it start? c)when is the body momentarily at rest? d) What is the acceleration of the body?

$${A}\:{small}\:{body}\:{is}\:{made}\:{to}\:{travel}\:{linearly}\:{t}\:{sconds}\:{after}\:{the}\: \\ $$$${start}.{Its}\:{distance}\:{S}\:{meters}\:{from}\:{a}\:{fix}\:{point}\:{O}\:{on}\:{a}\: \\ $$$${linear}\:{scale}\:{is}\:{given}\:{by}\:{S}\:=\:{t}^{\mathrm{2}} −\mathrm{5}{t}\:+\:\mathrm{6}. \\ $$$$\left.{a}\right)\:{How}\:{far}\:{is}\:{the}\:{body}\:{from}\:{O}\:{at}\:{the}\:{start}? \\ $$$$\left.{b}\right){with}\:{what}\:{velocity}\:{does}\:{it}\:{start}? \\ $$$$\left.{c}\right){when}\:{is}\:{the}\:{body}\:{momentarily}\:{at}\:{rest}? \\ $$$$\left.{d}\right)\:{What}\:{is}\:{the}\:{acceleration}\:{of}\:{the}\:{body}? \\ $$

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