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Question Number 34850    Answers: 2   Comments: 0

Question Number 34849    Answers: 0   Comments: 2

let f(x) = (e^(−x) /(2+x)) developp f at integr serie.

$${let}\:{f}\left({x}\right)\:=\:\:\:\:\frac{{e}^{−{x}} }{\mathrm{2}+{x}} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 34843    Answers: 0   Comments: 1

lim_ _(x→∞) ((ln x)/x) = ? You can only use series expansion / sandwich theorem!

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}_{} }\:\frac{\mathrm{ln}\:{x}}{{x}}\:=\:? \\ $$$${You}\:{can}\:\:{only}\:{use} \\ $$$${series}\:{expansion}\:/\:{sandwich}\:{theorem}! \\ $$

Question Number 34827    Answers: 1   Comments: 5

Find ∫ Sin^6 x dx

$$\boldsymbol{{Find}}\:\int\:\boldsymbol{{Sin}}^{\mathrm{6}} \boldsymbol{{x}}\:\boldsymbol{{dx}} \\ $$$$ \\ $$

Question Number 34821    Answers: 2   Comments: 1

Find range of y=(x/((x−1)(x−2))) .

$${Find}\:{range}\:{of} \\ $$$$\:\:\:{y}=\frac{{x}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)}\:. \\ $$

Question Number 34792    Answers: 2   Comments: 0

solve for x 4x=2^x

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{x}} \\ $$$$\mathrm{4}\boldsymbol{{x}}=\mathrm{2}^{\boldsymbol{{x}}} \\ $$

Question Number 34845    Answers: 0   Comments: 0

f(x)=cos(x) g(x)=2^x f:R→R g:R→R^+ [f(x)]^2 +[f(π/2−x)]^2 =((log_2 g(x))/x);x≠0 f(g(x)x)=[f(g(x−1)x)]^2 +[f(π/2−g(x−1)x)]^2 find f and g

$${f}\left({x}\right)=\mathrm{cos}\left({x}\right) \\ $$$${g}\left({x}\right)=\mathrm{2}^{{x}} \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${g}:\mathbb{R}\rightarrow\mathbb{R}^{+} \\ $$$$\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{f}\left(\pi/\mathrm{2}−{x}\right)\right]^{\mathrm{2}} =\frac{\mathrm{log}_{\mathrm{2}} {g}\left({x}\right)}{{x}};{x}\neq\mathrm{0} \\ $$$${f}\left({g}\left({x}\right){x}\right)=\left[{f}\left({g}\left({x}−\mathrm{1}\right){x}\right)\right]^{\mathrm{2}} +\left[{f}\left(\pi/\mathrm{2}−{g}\left({x}−\mathrm{1}\right){x}\right)\right]^{\mathrm{2}} \\ $$$$\mathrm{find}\:{f}\:\mathrm{and}\:{g} \\ $$

Question Number 34803    Answers: 1   Comments: 2

Question Number 34781    Answers: 1   Comments: 0

Question Number 34774    Answers: 1   Comments: 1

find lim_(n→+∞) ((n^p sin^2 (n!))/n^(p+1) ) with0<p<1 .

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{n}^{{p}} \:{sin}^{\mathrm{2}} \left({n}!\right)}{{n}^{{p}+\mathrm{1}} }\:\:{with}\mathrm{0}<{p}<\mathrm{1}\:. \\ $$

Question Number 34771    Answers: 0   Comments: 1

let A(x)= ∫_0 ^1 ln(1+ix^2 )dx find a simple form of f(x) (x∈R)

$${let}\:{A}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{ix}^{\mathrm{2}} \right){dx} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right)\:\:\:\:\left({x}\in{R}\right) \\ $$

Question Number 34770    Answers: 1   Comments: 2

let f(x)= ln(1+ix^2 ) 1) extrsct Re(f(x)) and Im(f(x)) 2) developp f at integr serie 3) calculate f^′ (x) by two methods

$${let}\:{f}\left({x}\right)=\:{ln}\left(\mathrm{1}+{ix}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{extrsct}\:{Re}\left({f}\left({x}\right)\right)\:{and}\:{Im}\left({f}\left({x}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:{by}\:{two}\:{methods} \\ $$

Question Number 34765    Answers: 1   Comments: 0

Question Number 34762    Answers: 1   Comments: 0

Question Number 34760    Answers: 2   Comments: 0

Question Number 34759    Answers: 1   Comments: 0

Question Number 34755    Answers: 1   Comments: 5

lim_(x→0) ((1/x) − ((ln^(1000) (1 + x))/x^(1001) ))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{{x}}\:−\:\frac{\mathrm{ln}^{\mathrm{1000}} \:\left(\mathrm{1}\:+\:{x}\right)}{{x}^{\mathrm{1001}} }\right) \\ $$

Question Number 34746    Answers: 0   Comments: 0

y+(d^2 y/dx^2 )=(1/x)(xln x−(dy/dx)+cos x) y=?

$${y}+\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\frac{\mathrm{1}}{{x}}\left({x}\mathrm{ln}\:{x}−\frac{{dy}}{{dx}}+\mathrm{cos}\:{x}\right) \\ $$$${y}=? \\ $$

Question Number 34733    Answers: 0   Comments: 0

Someone plz solve Q 34652

$${Someone}\:{plz}\:{solve}\: \\ $$$${Q}\:\mathrm{34652}\: \\ $$

Question Number 34724    Answers: 1   Comments: 1

use bernoulli methods sec^2 y(dy/dx)+xtany=x^3

$$\boldsymbol{\mathrm{use}}\:\boldsymbol{\mathrm{bernoulli}}\:\boldsymbol{\mathrm{methods}} \\ $$$$\boldsymbol{\mathrm{sec}}^{\mathrm{2}} \boldsymbol{{y}}\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}+\boldsymbol{{x}\mathrm{tan}{y}}=\boldsymbol{{x}}^{\mathrm{3}} \\ $$

Question Number 34739    Answers: 3   Comments: 3

Question Number 34738    Answers: 0   Comments: 2

Question Number 34737    Answers: 0   Comments: 0

Question Number 34736    Answers: 0   Comments: 1

letf(x)=−2x +(√(x−3)) 1) find f^(−1) (x) 2) calculate (f^(−1) )^′ (x) and (f^(−1) )^, (2) 3) let g(x) = x^2 −2x+3 calculate fog(x) and give D_(fog) 4) find (fog)^(−1) (x) 5) calculate ((fog)^(−1) )^′ (x).

$${letf}\left({x}\right)=−\mathrm{2}{x}\:\:+\sqrt{{x}−\mathrm{3}} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\left({f}^{−\mathrm{1}} \right)^{'} \left({x}\right)\:\:\:{and}\:\left({f}^{−\mathrm{1}} \right)^{,} \left(\mathrm{2}\right) \\ $$$$\left.\mathrm{3}\right)\:{let}\:{g}\left({x}\right)\:=\:{x}^{\mathrm{2}} \:−\mathrm{2}{x}+\mathrm{3} \\ $$$${calculate}\:{fog}\left({x}\right)\:{and}\:{give}\:{D}_{{fog}} \\ $$$$\left.\mathrm{4}\right)\:{find}\:\left({fog}\right)^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:\:\left(\left({fog}\right)^{−\mathrm{1}} \right)^{'} \left({x}\right). \\ $$

Question Number 34734    Answers: 1   Comments: 1

given the sum of the first n terms of an AP is x^2 the sum of the first 2n terms of the same AP is x^2 +x show that the sum of the first 4n terms is 4x^2 −8x+4

$$\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\: \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{AP}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{{x}}^{\mathrm{2}} \:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{AP}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{x}} \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\mathrm{4}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\:\mathrm{is} \\ $$$$\mathrm{4}\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{8}\boldsymbol{{x}}+\mathrm{4} \\ $$

Question Number 34721    Answers: 0   Comments: 0

let ξ(x) = Σ_(n=1) ^∞ (1/n^x ) with x>1 1)prove that (1/(x−1)) ≤ξ(x)≤ (x/(x−1)) 2) find lim_(x→1^+ ) (x−1)ξ(x) .

$${let}\:\xi\left({x}\right)\:=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:{with}\:{x}>\mathrm{1} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:\frac{\mathrm{1}}{{x}−\mathrm{1}}\:\leqslant\xi\left({x}\right)\leqslant\:\frac{{x}}{{x}−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow\mathrm{1}^{+} } \left({x}−\mathrm{1}\right)\xi\left({x}\right)\:. \\ $$

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