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AllQuestion and Answers: Page 1559

Question Number 42559    Answers: 5   Comments: 0

Question Number 42553    Answers: 0   Comments: 1

Question Number 42550    Answers: 1   Comments: 3

2^(1/x) =(√x) Find x

$$\mathrm{2}^{\frac{\mathrm{1}}{{x}}} =\sqrt{{x}} \\ $$$${Find}\:{x} \\ $$

Question Number 43535    Answers: 2   Comments: 2

1) find the value of ∫_(π/4) ^(π/3) (√(1+tanθ))dθ .

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\sqrt{\mathrm{1}+{tan}\theta}{d}\theta\:. \\ $$

Question Number 42549    Answers: 1   Comments: 0

If y = a^(1/(1 − log_a x)) and z = a^(1/(1 − log_a y)) . show that x = a^(1/(1 − log_a z))

$$\mathrm{If}\:\:\:\mathrm{y}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{x}}} \:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{y}}} \:\:.\:\:\mathrm{show}\:\mathrm{that}\:\:\:\:\:\mathrm{x}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{z}}} \\ $$

Question Number 42543    Answers: 0   Comments: 3

Question Number 42538    Answers: 1   Comments: 1

Question Number 43537    Answers: 0   Comments: 1

let S_n =Σ_(k=1) ^n (e^(k/n) /n) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{e}^{\frac{{k}}{{n}}} }{{n}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {S}_{{n}} \\ $$

Question Number 42521    Answers: 1   Comments: 0

Let a^(→ ) , b^→ , c^→ be three unit vectors such that 3a^→ +4b^→ +5c^→ = 0. Then prove that a^(→ ) , b^→ ,c^→ are coplanar.

$$\mathrm{Let}\:\overset{\rightarrow\:} {\mathrm{a}},\:\overset{\rightarrow} {\mathrm{b}}\:,\:\overset{\rightarrow} {\mathrm{c}}\:\mathrm{be}\:\mathrm{three}\:\mathrm{unit}\:\mathrm{vectors} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{3}\overset{\rightarrow} {\mathrm{a}}+\mathrm{4}\overset{\rightarrow} {\mathrm{b}}+\mathrm{5}\overset{\rightarrow} {\mathrm{c}}\:=\:\mathrm{0}.\:\mathrm{Then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\overset{\rightarrow\:} {\mathrm{a}},\:\overset{\rightarrow} {\mathrm{b}},\overset{\rightarrow} {\mathrm{c}}\:\mathrm{are}\:\mathrm{coplanar}. \\ $$

Question Number 42520    Answers: 1   Comments: 0

cos^3 A.sin3A+sinA.cos3A=(3/4)sin4A

$${cos}^{\mathrm{3}} \:{A}.{sin}\mathrm{3}{A}+{sinA}.{cos}\mathrm{3}{A}=\frac{\mathrm{3}}{\mathrm{4}}{sin}\mathrm{4}{A} \\ $$

Question Number 42519    Answers: 1   Comments: 0

cosec2A+cosec4A+cosec8A=cotA−cot8A(prlve ghis)

$${cosec}\mathrm{2}{A}+{cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}={cotA}−{cot}\mathrm{8}{A}\left({prlve}\:{ghis}\right) \\ $$

Question Number 42516    Answers: 1   Comments: 1

(x−(1/2)a),(x−(5/2)a)=? Solve Please.

$$\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{a}\right),\left(\mathrm{x}−\frac{\mathrm{5}}{\mathrm{2}}\mathrm{a}\right)=? \\ $$$$\mathrm{Solve}\:\mathrm{Please}. \\ $$

Question Number 42514    Answers: 1   Comments: 0

calculate d(x!)/dx= ?

$${calculate}\:\:{d}\left({x}!\right)/{dx}=\:? \\ $$

Question Number 42508    Answers: 2   Comments: 0

find the value of A =cos((π/5)).cos(((2π)/5)) cos(((4π)/5)) and B =sin((π/5))sin(((2π)/5))sin(((4π)/5)).

$${find}\:{the}\:{value}\:{of}\: \\ $$$${A}\:={cos}\left(\frac{\pi}{\mathrm{5}}\right).{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:{and}\:{B}\:={sin}\left(\frac{\pi}{\mathrm{5}}\right){sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right){sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right). \\ $$

Question Number 42507    Answers: 0   Comments: 2

let j =e^((i2π)/3) and p(x) =(1+xj)^n −(1−xj)^n 1) find the roots of p(x) and factorize inside C[x] p(x) 2) decompose inside C(x) the fration F(x) =(1/(p(x))) .

$${let}\:{j}\:={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:\:\:{p}\left({x}\right)\:=\left(\mathrm{1}+{xj}\right)^{{n}} \:−\left(\mathrm{1}−{xj}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:{and}\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fration}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{p}\left({x}\right)}\:. \\ $$

Question Number 42506    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ ((ln(t))/((^3 (√t^2 ))(1+t)))dt .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({t}\right)}{\left(^{\mathrm{3}} \sqrt{{t}^{\mathrm{2}} }\right)\left(\mathrm{1}+{t}\right)}{dt}\:. \\ $$

Question Number 42505    Answers: 0   Comments: 2

calculate ∫_0 ^∞ ((ln(t))/((1+t)(√t))) dt .

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{{t}}}\:{dt}\:. \\ $$

Question Number 42504    Answers: 0   Comments: 1

let x>0 prove that ∫_0 ^∞ ((e^(−t^2 ) ln(1+xt^2 ))/t^2 ) dt =π ∫_0 ^(√x) e^(1/u^2 ) du .

$${let}\:{x}>\mathrm{0}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{t}^{\mathrm{2}} } {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }\:{dt}\:=\pi\:\int_{\mathrm{0}} ^{\sqrt{{x}}} \:\:{e}^{\frac{\mathrm{1}}{{u}^{\mathrm{2}} }} \:\:{du}\:. \\ $$

Question Number 42503    Answers: 1   Comments: 1

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

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

Question Number 42502    Answers: 0   Comments: 0

calculate A_n = ∫_0 ^(2π) ((cos(nx))/(cosx +sinx))dx with n from N

$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\left({nx}\right)}{{cosx}\:+{sinx}}{dx}\:\:{with}\:{n}\:{from}\:{N} \\ $$

Question Number 42501    Answers: 0   Comments: 1

calculate I = ∫_0 ^(2π) ((cos(4x))/(cosx +sinx)) and J = ∫_0 ^(2π) ((sin(4x))/(cosx +sinx))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{4}{x}\right)}{{cosx}\:+{sinx}}\:\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{4}{x}\right)}{{cosx}\:+{sinx}}{dx} \\ $$

Question Number 42500    Answers: 0   Comments: 2

calculate I = ∫_0 ^(2π) ((cos(2x))/(cosx +sinx))dx and J =∫_0 ^(2π) ((sin(2x))/(cosx +sinx))dx

$${calculate}\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{{cosx}\:+{sinx}}{dx}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{cosx}\:+{sinx}}{dx} \\ $$

Question Number 42497    Answers: 0   Comments: 1

A body cools from 90°C to 40°C in 2 minutes at a temperature,20°C of the surrounding.Calculate the temperature of the body after another 5 minutes.

$${A}\:{body}\:{cools}\:{from}\:\mathrm{90}°{C}\:{to}\:\mathrm{40}°{C}\:{in} \\ $$$$\mathrm{2}\:{minutes}\:{at}\:{a}\:{temperature},\mathrm{20}°{C} \\ $$$${of}\:{the}\:{surrounding}.{Calculate}\:{the} \\ $$$${temperature}\:{of}\:{the}\:{body}\:{after} \\ $$$${another}\:\mathrm{5}\:{minutes}. \\ $$

Question Number 42495    Answers: 0   Comments: 0

In the sequence 1, 22, 333, ... 10101010101010101010, 1111111111111111111111, ... The sum of the digits in the 200th term is ??

$$\mathrm{In}\:\mathrm{the}\:\mathrm{sequence}\:\:\mathrm{1},\:\mathrm{22},\:\mathrm{333},\:...\:\mathrm{10101010101010101010},\:\mathrm{1111111111111111111111},\:... \\ $$$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{the}\:\mathrm{200th}\:\mathrm{term}\:\mathrm{is}\:?? \\ $$

Question Number 42494    Answers: 1   Comments: 1

In the sequence of numbers 1, 2, 11, 22, 111, 222, ... the sum of the digits in 999th terms is ??

$$\mathrm{In}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{numbers}\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{11},\:\mathrm{22},\:\mathrm{111},\:\mathrm{222},\:...\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{in}\:\mathrm{999th}\:\mathrm{terms}\:\mathrm{is}\:?? \\ $$

Question Number 42493    Answers: 0   Comments: 1

calculate lim_(n→+∞) Σ_(1≤i<j≤n) (1/(i^x j^x )) with x>1 for that consider ξ(x) =Σ_(n=1) ^∞ (1/n^x ) 2) calculate lim_(n→+∞) Σ_(1≤i<j≤n) (1/((ij)^2 )) .

$$\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\:\:\frac{\mathrm{1}}{{i}^{{x}} {j}^{{x}} }\:\:\:{with}\:\:{x}>\mathrm{1}\:\:{for}\:{that}\:{consider} \\ $$$$\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{{x}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\:\:\:\frac{\mathrm{1}}{\left({ij}\right)^{\mathrm{2}} }\:. \\ $$

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