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AllQuestion and Answers: Page 1436

Question Number 65153 Answers: 1 Comments: 2

Question Number 65148 Answers: 0 Comments: 3

Question Number 65137 Answers: 0 Comments: 3

∫(dx/((x−2)^3 (x+1)^2 ))=?

$$\int\frac{{dx}}{\left({x}−\mathrm{2}\right)^{\mathrm{3}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }=? \\ $$

Question Number 65134 Answers: 0 Comments: 4

let U_n a sequence U_0 =a and U_n =nU_(n−1) −2 (n>0) calculate U_n interms of n.

$${let}\:{U}_{{n}} \:{a}\:{sequence}\:{U}_{\mathrm{0}} ={a}\:{and} \\ $$$${U}_{{n}} ={nU}_{{n}−\mathrm{1}} \:\:\:−\mathrm{2}\:\:\:\left({n}>\mathrm{0}\right) \\ $$$${calculate}\:{U}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 65133 Answers: 0 Comments: 1

find f(x)=∫_0 ^(π/4) ln(sint +xcost)dt x real.

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left({sint}\:+{xcost}\right){dt} \\ $$$${x}\:{real}. \\ $$

Question Number 65132 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((tarctan(2t))/(1+t^4 ))dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{tarctan}\left(\mathrm{2}{t}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt} \\ $$

Question Number 65131 Answers: 0 Comments: 1

calculate ∫_0 ^∞ (dx/((x^4 −4i)^3 ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} −\mathrm{4}{i}\right)^{\mathrm{3}} } \\ $$

Question Number 65130 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((x^2 −3)/((x^4 +x^2 +2)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{2}} −\mathrm{3}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 65129 Answers: 1 Comments: 1

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

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

Question Number 65128 Answers: 2 Comments: 0

Question Number 65122 Answers: 0 Comments: 0

Question Number 65119 Answers: 0 Comments: 3

Question Number 65102 Answers: 1 Comments: 1

The polynomial 5x^5 −3x^3 +2x^2 −k gives a remainder 1, when divided by x+1. Find the value of k.

$$\mathrm{The}\:\mathrm{polynomial}\:\:\:\mathrm{5}{x}^{\mathrm{5}} −\mathrm{3}{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{2}} −{k}\: \\ $$$$\mathrm{gives}\:\mathrm{a}\:\mathrm{remainder}\:\mathrm{1},\:\mathrm{when}\:\mathrm{divided} \\ $$$$\mathrm{by}\:{x}+\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}. \\ $$

Question Number 65100 Answers: 0 Comments: 8

∫_0 ^π (dθ/((a+cosθ)^2 )), a>1

$$\int_{\mathrm{0}} ^{\pi} \frac{{d}\theta}{\left({a}+{cos}\theta\right)^{\mathrm{2}} },\:{a}>\mathrm{1} \\ $$

Question Number 65092 Answers: 0 Comments: 2

calculate ∫ (1/(x cosx))Π_(i=1) ^n (1−tan^2 ((x/2^i )))dx

$${calculate}\:\:\int\:\:\frac{\mathrm{1}}{{x}\:{cosx}}\prod_{{i}=\mathrm{1}} ^{{n}} \left(\mathrm{1}−{tan}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}^{{i}} }\right)\right){dx} \\ $$

Question Number 65087 Answers: 1 Comments: 0

Question Number 65077 Answers: 1 Comments: 0

Question Number 65115 Answers: 1 Comments: 0

x^x =64 find x

$$\mathrm{x}^{\mathrm{x}} =\mathrm{64} \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$

Question Number 65114 Answers: 0 Comments: 0

x^x^(lnx) =64 find x

$$\mathrm{x}^{\mathrm{x}^{\mathrm{lnx}} } =\mathrm{64} \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$

Question Number 65113 Answers: 0 Comments: 1

Question Number 65062 Answers: 1 Comments: 5

If x^4 +ax^2 +bx+c=0 ⇒ t^4 +At^2 +B=0 Find A and B.

$${If}\:\:{x}^{\mathrm{4}} +{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$$\Rightarrow\:{t}^{\mathrm{4}} +{At}^{\mathrm{2}} +{B}=\mathrm{0} \\ $$$${Find}\:{A}\:{and}\:{B}. \\ $$

Question Number 65061 Answers: 0 Comments: 2

let f(x) =∫_0 ^∞ (dt/((x−t +t^2 )^3 )) with x>(1/4) 1) calculate f(x) 2) calculate also g(x) =∫_0 ^∞ (dt/((x−t+t^2 )^4 )) 3)find the values of ∫_0 ^∞ (dt/((1−t+t^2 )^3 )) and ∫_0 ^∞ (dt/((2−t+t^2 )^4 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({x}−{t}\:+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{with}\:\:\:{x}>\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{also}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dt}}{\left({x}−{t}+{t}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left(\mathrm{1}−{t}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left(\mathrm{2}−{t}+{t}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$ \\ $$

Question Number 65059 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) (dx/((x^2 −x+1)^4 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 65054 Answers: 2 Comments: 2

{ (((√(x+y))+(√(x−y))=a)),((x^2 +y^2 =b [a,b∈R])) :}

$$\begin{cases}{\sqrt{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}}+\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}}=\boldsymbol{\mathrm{a}}}\\{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\boldsymbol{\mathrm{b}}\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}}\in\boldsymbol{\mathrm{R}}\right]}\end{cases} \\ $$

Question Number 65052 Answers: 4 Comments: 0

A.Evaluate: (i)∫((sin x+cos x)/(9+16sin 2x))dx (ii)∫((1+x^2 )/((1−x^2 )(√(1+x^2 +x^4 ))))dx (iii)∫((x−1)/((x+1)(√(x^3 +x+x^2 ))))dx

$${A}.\mathrm{Evaluate}: \\ $$$$\left(\mathrm{i}\right)\int\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{9}+\mathrm{16sin}\:\mathrm{2}{x}}{dx} \\ $$$$\left(\mathrm{ii}\right)\int\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} }}{dx} \\ $$$$\left(\mathrm{iii}\right)\int\frac{{x}−\mathrm{1}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{3}} +{x}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 65044 Answers: 0 Comments: 1

solve x^2 y^(′′) +xy^′ +y =0 on ]0,+∞[ (put x =e^t )

$$\left.{solve}\:{x}^{\mathrm{2}} {y}^{''} \:+{xy}^{'} \:+{y}\:=\mathrm{0}\:\:{on}\:\right]\mathrm{0},+\infty\left[\:\:\:\left({put}\:{x}\:={e}^{{t}} \right)\right. \\ $$

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