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AllQuestion and Answers: Page 1345

Question Number 78623 Answers: 1 Comments: 0

calculate lim_(x→0) (((√(1+x+x^2 +....+x^n )) −1)/x^(n/2) )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{\sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} +....+{x}^{{n}} }\:\:−\mathrm{1}}{{x}^{\frac{{n}}{\mathrm{2}}} } \\ $$

Question Number 78622 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^n (1−(t/n))^n ln(1+nt)dt

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:\:\int_{\mathrm{0}} ^{{n}} \left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} {ln}\left(\mathrm{1}+{nt}\right){dt} \\ $$

Question Number 78621 Answers: 0 Comments: 1

calculate lim_(x→1) ∫_x ^x^3 ((sh(xt^2 ))/(sin(xt)))dt

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\:\:\int_{{x}} ^{{x}^{\mathrm{3}} } \:\:\frac{{sh}\left({xt}^{\mathrm{2}} \right)}{{sin}\left({xt}\right)}{dt} \\ $$

Question Number 78620 Answers: 1 Comments: 0

explicit f(x) =∫_0 ^(+∞) ln(1−xe^(−t) )dt with ∣x∣<1

$${explicit}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} {ln}\left(\mathrm{1}−{xe}^{−{t}} \right){dt}\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$

Question Number 78609 Answers: 0 Comments: 0

Question Number 78628 Answers: 2 Comments: 1

Find minimum value of y = ((2x)/(x^2 + x + 1)) x , y ∈ R Without Differential

$${Find}\:\:{minimum}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:{y}\:\:=\:\:\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}} \\ $$$${x}\:,\:{y}\:\:\in\:\:\mathbb{R} \\ $$$${Without}\:\:{Differential} \\ $$

Question Number 78596 Answers: 1 Comments: 1

lim_(x→0) (1−3tan^2 x)^((2/(sin^2 3x)) ) = ?

$$ \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{1}−\mathrm{3tan}\:^{\mathrm{2}} \mathrm{x}\right)^{\frac{\mathrm{2}}{\mathrm{sin}\:^{\mathrm{2}} \:\mathrm{3x}}\:\:\:} =\:?\: \\ $$

Question Number 78581 Answers: 0 Comments: 4

∫ ((2xsin 2x)/((2x−sin 2x)^2 )) dx ?

$$\int\:\frac{\mathrm{2}{x}\mathrm{sin}\:\mathrm{2}{x}}{\left(\mathrm{2}{x}−\mathrm{sin}\:\mathrm{2}{x}\right)^{\mathrm{2}} }\:{dx}\:? \\ $$

Question Number 78578 Answers: 0 Comments: 0

Question Number 78575 Answers: 1 Comments: 0

∫ (√(tan x)) dx

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

Question Number 78569 Answers: 1 Comments: 0

which of the following is increasing or decreasing a. u_n = ((n!)/n^n ) b. u_n = (4^n /(3^n +1)) c. u_n = (2^n /n^2 )

$${which}\:{of}\:{the}\:{following}\:{is}\:{increasing}\:{or}\:{decreasing} \\ $$$${a}.\:\:{u}_{{n}} \:=\:\frac{{n}!}{{n}^{{n}} } \\ $$$${b}.\:\:{u}_{{n}} =\:\frac{\mathrm{4}^{{n}} }{\mathrm{3}^{{n}} +\mathrm{1}} \\ $$$${c}.\:{u}_{{n}} =\:\frac{\mathrm{2}^{{n}} }{{n}^{\mathrm{2}} } \\ $$

Question Number 78568 Answers: 1 Comments: 0

Find the roots of the equation bx^3 − (3b + 2)x^2 − 2(5b − 3)x + 20 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\mathrm{bx}^{\mathrm{3}} \:−\:\left(\mathrm{3b}\:+\:\mathrm{2}\right)\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{2}\left(\mathrm{5b}\:−\:\mathrm{3}\right)\mathrm{x}\:+\:\mathrm{20}\:\:=\:\:\mathrm{0} \\ $$

Question Number 78567 Answers: 0 Comments: 9

Question Number 78564 Answers: 1 Comments: 1

Question Number 78563 Answers: 0 Comments: 0

Question Number 78549 Answers: 1 Comments: 0

Question Number 78542 Answers: 0 Comments: 0

Question Number 78526 Answers: 0 Comments: 14

lim_(x→0) [((∫_( 0) ^( x^2 ) (√(4 + t^3 )) dt)/x^2 )]

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left[\frac{\int_{\:\:\mathrm{0}} ^{\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\sqrt{\mathrm{4}\:+\:\boldsymbol{\mathrm{t}}^{\mathrm{3}} }\:\:\boldsymbol{\mathrm{dt}}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right] \\ $$

Question Number 78522 Answers: 0 Comments: 10

what is the line passing through (2,2,1) and parallel to 2i^ − j^ − k^ ?

$${what}\:{is}\:{the}\: \\ $$$${line}\:{passing}\:{through}\:\left(\mathrm{2},\mathrm{2},\mathrm{1}\right) \\ $$$${and}\:{parallel}\:{to}\:\mathrm{2}\hat {{i}}\:−\:\hat {{j}}\:−\:\hat {{k}}\:? \\ $$

Question Number 78503 Answers: 3 Comments: 0

if x,y >1 prove (x^2 /(y−1))+(y^2 /(x−1))≥8

$${if}\:{x},{y}\:>\mathrm{1}\: \\ $$$${prove}\:\frac{{x}^{\mathrm{2}} }{{y}−\mathrm{1}}+\frac{{y}^{\mathrm{2}} }{{x}−\mathrm{1}}\geqslant\mathrm{8} \\ $$

Question Number 78496 Answers: 1 Comments: 4

Question Number 78493 Answers: 2 Comments: 0

the sum to infinity of a Geometric series is S the sum to infinty of the squares of the terms of the series is 2S the sum to infinity of the cubes of the terms of the series is ((64)/(13))S. find the value of S and write iut the first 3 terms if the series.

$${the}\:{sum}\:{to}\:{infinity}\:{of}\:{a}\:{Geometric}\:{series}\:{is}\:{S} \\ $$$${the}\:{sum}\:{to}\:{infinty}\:{of}\:{the}\:{squares}\:{of}\:{the}\:{terms} \\ $$$${of}\:{the}\:{series}\:{is}\:\mathrm{2}{S} \\ $$$${the}\:{sum}\:{to}\:{infinity}\:{of}\:{the}\:{cubes}\:{of}\:{the}\:{terms} \\ $$$${of}\:{the}\:{series}\:{is}\:\frac{\mathrm{64}}{\mathrm{13}}{S}. \\ $$$${find}\:{the}\:{value}\:{of}\:{S}\:{and}\:{write}\:{iut}\:{the}\:{first} \\ $$$$\mathrm{3}\:{terms}\:{if}\:{the}\:{series}. \\ $$

Question Number 78490 Answers: 1 Comments: 0

Find out A =Σ_(n=2) ^∞ ((ζ(n))/(n(−3)^n )) where ζ(p)=Σ_(n=1) ^∞ (1/n^p )

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{A}\:=\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\:\frac{\zeta\left(\mathrm{n}\right)}{\mathrm{n}\left(−\mathrm{3}\right)^{\mathrm{n}} }\:\:\:\:\:\:\:\:\:\:\:\mathrm{where}\:\:\zeta\left(\mathrm{p}\right)=\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{p}} }\: \\ $$

Question Number 78489 Answers: 1 Comments: 0

let P(x)= x^5 −209x+56 Prove that there exist two roots a,b such as ab=1 Find out their sum ( a+b=?) and deduce the decomposition of P(x) in prime factors.

$$\mathrm{let}\:\:\mathrm{P}\left(\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{5}} −\mathrm{209x}+\mathrm{56}\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{two}\:\mathrm{roots}\:\:\mathrm{a},\mathrm{b}\:\mathrm{such}\:\mathrm{as}\:\:\:\mathrm{ab}=\mathrm{1} \\ $$$$\mathrm{Find}\:\mathrm{out}\:\mathrm{their}\:\mathrm{sum}\:\left(\:\mathrm{a}+\mathrm{b}=?\right)\:\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{decomposition}\:\mathrm{of}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{in}\:\mathrm{prime}\:\mathrm{factors}. \\ $$

Question Number 78511 Answers: 1 Comments: 2

Question Number 78468 Answers: 0 Comments: 1

If F(x) = ∫_0 ^(arctanx) (√(((t^4 −1)/(t^4 +1))dt,)) find F′(x).

$${If}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{{arctanx}} \sqrt{\frac{{t}^{\mathrm{4}} −\mathrm{1}}{{t}^{\mathrm{4}} +\mathrm{1}}{dt},}\:{find}\:{F}'\left({x}\right). \\ $$

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