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Question Number 27805    Answers: 0   Comments: 1

find ∫_1 ^∝ ((arctan(αx))/x^2 ) .

$${find}\:\:\int_{\mathrm{1}} ^{\propto} \:\:\frac{{arctan}\left(\alpha{x}\right)}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 27804    Answers: 0   Comments: 1

calculate ∫_0 ^∝ ((e^(−ax) − e^(−bx) )/x^2 )dx with a>0 b>o

$${calculate}\:\:\int_{\mathrm{0}} ^{\propto} \:\:\frac{{e}^{−{ax}} \:−\:{e}^{−{bx}} }{{x}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$${b}>{o} \\ $$

Question Number 27803    Answers: 0   Comments: 0

find the value of ∫_0 ^1 ((arctan(x +x^(−1) ))/(1+x^2 )) dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({x}\:+{x}^{−\mathrm{1}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 27802    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ (e^(−2x^2 ) /((3+x^2 )^2 ))dx .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{2}{x}^{\mathrm{2}} } }{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 27801    Answers: 0   Comments: 1

solve the d.e. y^′ −2xy =sinx e^x^2 with initial condition y(o)=1.

$${solve}\:{the}\:{d}.{e}.\:\:{y}^{'} \:−\mathrm{2}{xy}\:={sinx}\:{e}^{{x}^{\mathrm{2}} } \:{with}\:{initial}\:{condition} \\ $$$${y}\left({o}\right)=\mathrm{1}. \\ $$

Question Number 27912    Answers: 0   Comments: 2

find the sum of Σ_(n=1) ^∝ (1/n)( ((√2)/(1+i)))^n .

$${find}\:{the}\:{sum}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\propto} \frac{\mathrm{1}}{{n}}\left(\:\:\frac{\sqrt{\mathrm{2}}}{\mathrm{1}+{i}}\right)^{{n}} . \\ $$

Question Number 27797    Answers: 0   Comments: 1

find ∫ (√(2+tan^2 t)) dt.

$${find}\:\:\:\int\:\sqrt{\mathrm{2}+{tan}^{\mathrm{2}} {t}}\:\:{dt}. \\ $$

Question Number 27796    Answers: 0   Comments: 0

find ∫ (x^2 /((cosx +x sinx)^2 )) .

$${find}\:\:\int\:\:\:\frac{{x}^{\mathrm{2}} }{\left({cosx}\:+{x}\:{sinx}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 27794    Answers: 0   Comments: 0

let give I(x)= ∫_0 ^π ln (1−2x cost +x^2 )dt by using the polynomial p(x)= (z+x)^(2n) −1 find the value of I(x).

$${let}\:{give}\:\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} {ln}\:\left(\mathrm{1}−\mathrm{2}{x}\:{cost}\:+{x}^{\mathrm{2}} \right){dt}\:{by}\:{using}\:{the} \\ $$$${polynomial}\:{p}\left({x}\right)=\:\left({z}+{x}\right)^{\mathrm{2}{n}} −\mathrm{1}\:\:{find}\:{the}\:{value}\:{of}\:{I}\left({x}\right). \\ $$

Question Number 27792    Answers: 0   Comments: 3

find lim_(x−>1) ∫_x ^x^2 (dt/(ln(t))) .

$${find}\:{lim}_{{x}−>\mathrm{1}} \int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\frac{{dt}}{{ln}\left({t}\right)}\:\:. \\ $$

Question Number 27790    Answers: 0   Comments: 2

let give f(x)= x^(n−1) ln(1+x) with n fromN^∗ find f^((n)) (x)

$${let}\:{give}\:\:{f}\left({x}\right)=\:{x}^{{n}−\mathrm{1}} \:{ln}\left(\mathrm{1}+{x}\right)\:\:{with}\:{n}\:{fromN}^{\ast} \:\:\:{find}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$

Question Number 27789    Answers: 0   Comments: 3

find lim_(x−>0) (1+sinx)^(1/x) .

$${find}\:\:{lim}_{{x}−>\mathrm{0}} \left(\mathrm{1}+{sinx}\right)^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 27788    Answers: 0   Comments: 0

find the value of A_n = ∫_0 ^π ((sin(nt))/(sint))dt with n∈N^∗ .

$${find}\:{the}\:{value}\:{of}\:\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}\left({nt}\right)}{{sint}}{dt}\:\:{with}\:{n}\in{N}^{\ast} \:. \\ $$

Question Number 27787    Answers: 1   Comments: 0

find lim_(x−>0) ((e^x −x−1)/x^2 ) .

$${find}\:\:{lim}_{{x}−>\mathrm{0}} \frac{{e}^{{x}} \:\:−{x}−\mathrm{1}}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 27786    Answers: 0   Comments: 1

find lim_(x−>0) (((tanx)/x))^(1/x^2 ) .

$${find}\:{lim}_{{x}−>\mathrm{0}} \:\left(\frac{{tanx}}{{x}}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} . \\ $$

Question Number 27785    Answers: 0   Comments: 1

find nature of the serie Σ_(n=1) ^∝ ξ(n) x^n with ξ(x)= Σ_(n=1) ^∝ (1/n^x ) and x>1 .

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\xi\left({n}\right)\:{x}^{{n}} \\ $$$${with}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{and}\:{x}>\mathrm{1}\:. \\ $$

Question Number 27784    Answers: 0   Comments: 0

let give f(x) = e^(−(1/x)) with f(0)=0 1) is f derivable in point 0? 2)prove that f^((n)) = F_n (x) e^(−(1/x)) with F_(n ) is rational function 3) calculate f^((6)) (x) and f^((9)) (x) .

$${let}\:{give}\:\:{f}\left({x}\right)\:=\:{e}^{−\frac{\mathrm{1}}{{x}}} \:\:\:{with}\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{is}\:{f}\:{derivable}\:{in}\:{point}\:\mathrm{0}? \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:{f}^{\left({n}\right)} =\:{F}_{{n}} \left({x}\right)\:{e}^{−\frac{\mathrm{1}}{{x}}} \:\:{with}\:{F}_{{n}\:} {is}\:{rational}\:{function} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\:{f}^{\left(\mathrm{6}\right)} \:\left({x}\right)\:{and}\:\:{f}^{\left(\mathrm{9}\right)} \left({x}\right)\:\:. \\ $$

Question Number 27782    Answers: 0   Comments: 1

solve the e.d. xy^′ +αy = xe^(−x) .

$${solve}\:{the}\:{e}.{d}.\:\:\:\:\:{xy}^{'} \:+\alpha{y}\:\:=\:{xe}^{−{x}} \:\:. \\ $$

Question Number 27781    Answers: 0   Comments: 1

find the value of F(x)=∫_0 ^(π/2) ((ln(1+x sin^2 t))/(sin^2 t)) dt knowing that −1<x<1 .

$${find}\:{the}\:{value}\:{of}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{x}\:{sin}^{\mathrm{2}} {t}\right)}{{sin}^{\mathrm{2}} {t}}\:{dt}\:{knowing}\:{that} \\ $$$$−\mathrm{1}<{x}<\mathrm{1}\:. \\ $$

Question Number 27774    Answers: 0   Comments: 1

Question Number 27771    Answers: 2   Comments: 1

Question Number 27769    Answers: 1   Comments: 0

A glass bottle full of mercury has mass 500g. On being heated through 35°C, 2.43g of mercury are expelled. calculate the mass of mercury remaining in the bottle (Cubic expansivity of mercury is 1.8 × 10^(−4) per K. linear expansivity of glass is 8.0 × 10^(−6) per K.

$$\mathrm{A}\:\mathrm{glass}\:\mathrm{bottle}\:\mathrm{full}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{has}\:\mathrm{mass}\:\mathrm{500g}.\:\mathrm{On}\:\mathrm{being}\:\mathrm{heated}\:\mathrm{through}\:\mathrm{35}°\mathrm{C}, \\ $$$$\mathrm{2}.\mathrm{43g}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{are}\:\mathrm{expelled}.\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{remaining}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{bottle}\:\:\left(\mathrm{Cubic}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{is}\:\mathrm{1}.\mathrm{8}\:×\:\mathrm{10}^{−\mathrm{4}} \:\mathrm{per}\:\mathrm{K}.\right. \\ $$$$\mathrm{linear}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{glass}\:\mathrm{is}\:\mathrm{8}.\mathrm{0}\:×\:\mathrm{10}^{−\mathrm{6}} \:\mathrm{per}\:\mathrm{K}. \\ $$

Question Number 27767    Answers: 1   Comments: 0

If the number of divisors of a number is odd,prove that the number is perfect square and vice versa.

$$\mathrm{If}\:\mathrm{the}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{divisors}}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{odd}},\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{is}\:\boldsymbol{\mathrm{perfect}}\:\boldsymbol{\mathrm{square}}\:\mathrm{and} \\ $$$$\mathrm{vice}\:\mathrm{versa}. \\ $$

Question Number 27764    Answers: 1   Comments: 0

∫(√(tan x))dx

$$\int\sqrt{\mathrm{tan}\:{x}}{dx} \\ $$

Question Number 27761    Answers: 1   Comments: 0

4(2a+b)^2 −(a−b)^2

$$\mathrm{4}\left(\mathrm{2a}+\mathrm{b}\right)^{\mathrm{2}} −\left(\mathrm{a}−\mathrm{b}\right)^{\mathrm{2}} \\ $$

Question Number 27729    Answers: 0   Comments: 0

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