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Question Number 105650    Answers: 2   Comments: 0

cos^(−1) (2x)+ cos^(−1) (x)=(π/6) find x

$$\mathrm{cos}^{−\mathrm{1}} \left(\mathrm{2}{x}\right)+\:\mathrm{cos}^{−\mathrm{1}} \left({x}\right)=\frac{\pi}{\mathrm{6}} \\ $$$${find}\:{x} \\ $$

Question Number 105646    Answers: 1   Comments: 0

Solve the differential equation; y′′−2ay′+(1+a^2 )y=te^(at) +sint

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}; \\ $$$$\mathrm{y}''−\mathrm{2ay}'+\left(\mathrm{1}+\mathrm{a}^{\mathrm{2}} \right)\mathrm{y}=\mathrm{te}^{\mathrm{at}} +\mathrm{sint} \\ $$

Question Number 105638    Answers: 2   Comments: 0

(d^2 y/dx^2 ) + 9y = cos 4x

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{9}{y}\:=\:\mathrm{cos}\:\mathrm{4}{x} \\ $$

Question Number 105634    Answers: 0   Comments: 0

Question Number 105632    Answers: 1   Comments: 0

Given I_n =∫_0 ^1 (((1−x)^n )/(n!))e^x dx , n∈N a\Show that ∀x∈[0,1], (1−x)^n e^x ≤e and deduce that the Sequence (I_n )_n converges to zero. b\Establish a recurrence relation between I_n and I_(n+1) c\ Deduce that e=lim_(n→∞) Σ_(k=0) ^n ((1/(k!)))

$$\mathrm{Given}\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{n}} }{\mathrm{n}!}\mathrm{e}^{\mathrm{x}} \mathrm{dx}\:,\:\mathrm{n}\in\mathbb{N} \\ $$$$\mathrm{a}\backslash\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{x}\in\left[\mathrm{0},\mathrm{1}\right],\:\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{n}} \mathrm{e}^{\mathrm{x}} \leqslant\mathrm{e}\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{Sequence}\:\left(\mathrm{I}_{\mathrm{n}} \right)_{\mathrm{n}} \:\mathrm{converges}\:\mathrm{to}\:\mathrm{zero}. \\ $$$$\mathrm{b}\backslash\mathrm{Establish}\:\mathrm{a}\:\mathrm{recurrence}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{I}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{I}_{\mathrm{n}+\mathrm{1}} \\ $$$$\mathrm{c}\backslash\:\mathrm{Deduce}\:\mathrm{that}\:\mathrm{e}=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{k}!}\right) \\ $$

Question Number 105630    Answers: 0   Comments: 2

Question Number 105625    Answers: 1   Comments: 0

prove that ((m),(m) ) ((( n)),((n−k)) )+ ((( m)),((m−1)) ) ((( n)),((n−k+1)) ) + ((( m)),((m−2)) ) ((( n)),((n−k+2)) )+......+ ((( m)),((m−k)) ) ((n),(n) ) = (((m+n)),(( k)) )

$${prove}\:{that} \\ $$$$\begin{pmatrix}{{m}}\\{{m}}\end{pmatrix}\begin{pmatrix}{\:\:\:{n}}\\{{n}−{k}}\end{pmatrix}+\begin{pmatrix}{\:\:\:\:{m}}\\{{m}−\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\:\:\:\:\:\:\:{n}}\\{{n}−{k}+\mathrm{1}}\end{pmatrix} \\ $$$$+\begin{pmatrix}{\:\:\:{m}}\\{{m}−\mathrm{2}}\end{pmatrix}\begin{pmatrix}{\:\:\:\:\:\:\:\:{n}}\\{{n}−{k}+\mathrm{2}}\end{pmatrix}+......+\begin{pmatrix}{\:\:\:\:{m}}\\{{m}−{k}}\end{pmatrix}\begin{pmatrix}{{n}}\\{{n}}\end{pmatrix} \\ $$$$=\begin{pmatrix}{{m}+{n}}\\{\:\:\:\:\:{k}}\end{pmatrix} \\ $$

Question Number 105619    Answers: 2   Comments: 0

find General solution cot x+cot 2x+cot3x= 0

$${find}\:\mathcal{G}{eneral}\:{solution}\:\mathrm{cot}\:{x}+\mathrm{cot}\:\mathrm{2}{x}+\mathrm{cot3}{x}=\:\mathrm{0} \\ $$$$ \\ $$

Question Number 105616    Answers: 2   Comments: 0

Question Number 105614    Answers: 2   Comments: 1

lim_(x→0) ((tan (cos 2x−1))/(2x^2 )) ?

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

Question Number 105613    Answers: 1   Comments: 0

f(x)=(1+(1/x))^(x!) f^′ (x)=????

$${f}\left({x}\right)=\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{x}!} \:\:\:\:\:\:\:\:\:\:\:\:{f}^{'} \left({x}\right)=???? \\ $$

Question Number 105609    Answers: 0   Comments: 0

Question Number 105605    Answers: 2   Comments: 0

prove by mathematical induction 2^3 +4^3 +6^3 +8^3 +...+(2n)^3 = 2n^2 (n+1)^2

$${prove}\:{by}\:{mathematical}\:{induction}\: \\ $$$$\mathrm{2}^{\mathrm{3}} +\mathrm{4}^{\mathrm{3}} +\mathrm{6}^{\mathrm{3}} +\mathrm{8}^{\mathrm{3}} +...+\left(\mathrm{2}{n}\right)^{\mathrm{3}} =\:\mathrm{2}{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} \\ $$

Question Number 105603    Answers: 2   Comments: 0

(d^2 y/dx^2 )−4(dy/dx)+y = a sin 2x

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{4}\frac{{dy}}{{dx}}+{y}\:=\:{a}\:\mathrm{sin}\:\mathrm{2}{x} \\ $$

Question Number 105599    Answers: 1   Comments: 0

Question Number 105598    Answers: 1   Comments: 0

Question Number 105596    Answers: 0   Comments: 1

Question Number 105594    Answers: 0   Comments: 3

Question Number 105593    Answers: 2   Comments: 0

f(x)=(((√(2x+1))×(√(2x+1)))/(2x+1)) Dom_f =?

$${f}\left({x}\right)=\frac{\sqrt{\mathrm{2}{x}+\mathrm{1}}×\sqrt{\mathrm{2}{x}+\mathrm{1}}}{\mathrm{2}{x}+\mathrm{1}}\:\:\:\:\:\:{Dom}_{{f}} =? \\ $$

Question Number 105584    Answers: 1   Comments: 0

lim_(x→0) [csc^2 (2x)−(1/(4x^2 )) ]?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\mathrm{csc}^{\mathrm{2}} \left(\mathrm{2}{x}\right)−\frac{\mathrm{1}}{\mathrm{4}{x}^{\mathrm{2}} }\:\right]? \\ $$

Question Number 105575    Answers: 0   Comments: 0

Question Number 105574    Answers: 1   Comments: 0

∫ ((x−1)/(x+x^2 ln x)) dx ?

$$\int\:\frac{{x}−\mathrm{1}}{{x}+{x}^{\mathrm{2}} \mathrm{ln}\:{x}}\:{dx}\:?\: \\ $$

Question Number 105571    Answers: 4   Comments: 0

Σ_(n = 1) ^∞ (n/((n+1)(n+2)(n+3))) =?

$$\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)}\:=? \\ $$

Question Number 105569    Answers: 1   Comments: 0

x^2 (dy/dx) −3xy−2y^2 = 0

$${x}^{\mathrm{2}} \:\frac{{dy}}{{dx}}\:−\mathrm{3}{xy}−\mathrm{2}{y}^{\mathrm{2}} \:=\:\mathrm{0}\: \\ $$

Question Number 105566    Answers: 0   Comments: 0

let f(x) =(x+1)^(2n) e^(−x) sin(x) 1) calculate f^((n)) (x)and f^((n)) (0) 2)developp f at integr serie 3) find ∫ f(x)dx

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2n}} \:\mathrm{e}^{−\mathrm{x}} \mathrm{sin}\left(\mathrm{x}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \:\left(\mathrm{x}\right)\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{find}\:\int\:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 105565    Answers: 1   Comments: 0

let f(x) =x^2 ln(1−x^3 ) 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie 3)calculate ∫ f(x)dx

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{x}^{\mathrm{2}} \mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$$$\left.\mathrm{3}\right)\mathrm{calculate}\:\int\:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

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