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AllQuestion and Answers: Page 1408

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 )

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

Question Number 65040 Answers: 0 Comments: 0

what is the curve of the curve caused by the earth to the space time _ fabric

$${what}\:{is}\:{the}\:{curve}\:{of}\:{the}\:{curve}\:{caused}\:\:{by} \\ $$$${the}\:{earth}\:{to}\:{the}\:{space}\:{time}\:\_\:{fabric} \\ $$

Question Number 65022 Answers: 1 Comments: 4

Question Number 65015 Answers: 1 Comments: 3

∫(((√(x+1)) − (√(x−1)))/((√(x+1)) + (√(x−1)))) dx

$$\int\frac{\sqrt{{x}+\mathrm{1}}\:−\:\sqrt{{x}−\mathrm{1}}}{\sqrt{{x}+\mathrm{1}}\:+\:\sqrt{{x}−\mathrm{1}}}\:{dx} \\ $$

Question Number 65013 Answers: 1 Comments: 0

why do we divide each term by n when given the question lim_(x→∞) ((3 +2n)/(1+n)) ?

$${why}\:{do}\:{we}\:{divide}\:{each}\:{term}\:{by}\:{n}\:{when}\:{given}\:{the}\:{question} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{3}\:+\mathrm{2}{n}}{\mathrm{1}+{n}}\:? \\ $$

Question Number 65011 Answers: 5 Comments: 1

1.(i)Evaluate:∫(1/(sin x−cos x+(√2)))dx (ii)Evaluate:∫2^2^2^x 2^2^x 2^x dx (iii)Evaluate:∫((cos^3 x)/(sin^2 x+sin x))dx 2.cosec [tan^(−1) {cos (cot^(−1) (sec(sin^(−1) a)))}]=What? 3.Prove that, sin [cot^(−1) {cos (tan^(−1) x)}]=(√((x^2 +1)/(x^2 +2))) 4.Mention Order and Degree and state also if it is linear or non-linear. y+(d^2 y/dx^2 )=((19)/(25))∫y^2 dx

$$\mathrm{1}.\left(\mathrm{i}\right)\mathrm{Evaluate}:\int\frac{\mathrm{1}}{\mathrm{sin}\:{x}−\mathrm{cos}\:{x}+\sqrt{\mathrm{2}}}{dx} \\ $$$$\left(\mathrm{ii}\right)\mathrm{Evaluate}:\int\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{{x}} } } \mathrm{2}^{\mathrm{2}^{{x}} } \mathrm{2}^{{x}} \:{dx} \\ $$$$\left(\mathrm{iii}\right)\mathrm{Evaluate}:\int\frac{\mathrm{cos}\:^{\mathrm{3}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{sin}\:{x}}{dx} \\ $$$$\mathrm{2}.\mathrm{cosec}\:\left[\mathrm{tan}^{−\mathrm{1}} \left\{\mathrm{cos}\:\left(\mathrm{cot}^{−\mathrm{1}} \left(\mathrm{sec}\left(\mathrm{sin}^{−\mathrm{1}} {a}\right)\right)\right)\right\}\right]=\mathrm{What}? \\ $$$$\mathrm{3}.\mathrm{Prove}\:\mathrm{that},\:\:\mathrm{sin}\:\left[\mathrm{cot}^{−\mathrm{1}} \left\{\mathrm{cos}\:\left(\mathrm{tan}^{−\mathrm{1}} {x}\right)\right\}\right]=\sqrt{\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{2}}} \\ $$$$\mathrm{4}.\mathrm{Mention}\:\mathrm{Order}\:\mathrm{and}\:\mathrm{Degree}\:\mathrm{and}\:\:\mathrm{state}\:\mathrm{also}\:\mathrm{if}\:\mathrm{it}\:\mathrm{is}\:\mathrm{linear}\:\mathrm{or}\:\mathrm{non}-{l}\mathrm{inear}. \\ $$$$\:\:\:\:\:{y}+\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{\mathrm{19}}{\mathrm{25}}\int{y}^{\mathrm{2}} \:{dx} \\ $$

Question Number 65004 Answers: 0 Comments: 1

let U_n = ∫_(1/n) ^(2/n) Γ(x)Γ(1−x)dx with n≥3 1) calculate and determine lim_(n→+∞) U_n 2) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\:\int_{\frac{\mathrm{1}}{{n}}} ^{\frac{\mathrm{2}}{{n}}} \:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right){dx}\:\:\:\:{with}\:{n}\geqslant\mathrm{3} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{and}\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

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