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Question Number 146778    Answers: 1   Comments: 0

Find the modulus of a complex number: Z = cos 40 + i sin 20 + 1 = ?

$${Find}\:{the}\:{modulus}\:{of}\:{a}\:{complex} \\ $$$${number}: \\ $$$${Z}\:=\:{cos}\:\mathrm{40}\:+\:{i}\:{sin}\:\mathrm{20}\:+\:\mathrm{1}\:=\:? \\ $$

Question Number 146776    Answers: 0   Comments: 2

Question Number 146775    Answers: 3   Comments: 0

Prove that ∫^( 𝛑) _( 0) tln(sint)dt= βˆ’(𝛑^2 /2)ln(2)

$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\:\:\underset{\:\mathrm{0}} {\int}^{\:\boldsymbol{\pi}} \boldsymbol{{tln}}\left(\boldsymbol{{sint}}\right)\boldsymbol{{dt}}=\:βˆ’\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{2}}\boldsymbol{{ln}}\left(\mathrm{2}\right) \\ $$

Question Number 146793    Answers: 1   Comments: 0

βˆ€nβ‰₯2, u_n =Ξ _(k=2) ^n cos ((Ο€/2^k )) et v_n =u_n sin ((Ο€/2^n )) convergence, nature, sens of variations and adjantes? u_n and v_n help me please

$$\forall{n}\geqslant\mathrm{2},\:{u}_{{n}} =\underset{{k}=\mathrm{2}} {\overset{{n}} {\prod}}\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}^{{k}} }\right)\:{et}\:{v}_{{n}} ={u}_{{n}} \mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}^{{n}} }\right) \\ $$$${convergence},\:{nature},\:{sens}\:{of}\:{variations}\:{and}\:{adjantes}? \\ $$$${u}_{{n}} \:{and}\:{v}_{{n}} \\ $$$${help}\:{me}\:{please} \\ $$

Question Number 146772    Answers: 1   Comments: 0

If z=cos ΞΈ+i sin ΞΈ, prove that cos^6 ΞΈ=(1/(32))(cos 6ΞΈ+6cos 4ΞΈ+15cos 2ΞΈ+10). Hence or otherwise, find the value of ∫_0 ^( a) (√((a^2 βˆ’x^2 )^5 )) dx.

$$\mathrm{If}\:{z}=\mathrm{cos}\:\theta+{i}\:\mathrm{sin}\:\theta,\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{cos}^{\mathrm{6}} \theta=\frac{\mathrm{1}}{\mathrm{32}}\left(\mathrm{cos}\:\mathrm{6}\theta+\mathrm{6cos}\:\mathrm{4}\theta+\mathrm{15cos}\:\mathrm{2}\theta+\mathrm{10}\right). \\ $$$$\mathrm{Hence}\:\mathrm{or}\:\mathrm{otherwise},\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:{a}} \sqrt{\left({a}^{\mathrm{2}} βˆ’{x}^{\mathrm{2}} \right)^{\mathrm{5}} }\:{dx}. \\ $$

Question Number 146771    Answers: 1   Comments: 0

Given that yβ€²β€²βˆ’4yβ€²+3y=0, y(0)=0, yβ€²(0)=2, find y(ln 2).

$$\mathrm{Given}\:\mathrm{that}\:{y}''βˆ’\mathrm{4}{y}'+\mathrm{3}{y}=\mathrm{0},\:{y}\left(\mathrm{0}\right)=\mathrm{0},\:{y}'\left(\mathrm{0}\right)=\mathrm{2}, \\ $$$$\mathrm{find}\:{y}\left(\mathrm{ln}\:\mathrm{2}\right). \\ $$

Question Number 146767    Answers: 1   Comments: 0

Ξ£_(n=0) ^∞ (((βˆ’1)^n )/(n+1))(1+(1/3)+...+(1/(2n+1)))=?

$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}+\mathrm{1}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+...+\frac{\mathrm{1}}{\mathrm{2n}+\mathrm{1}}\right)=? \\ $$

Question Number 146764    Answers: 0   Comments: 0

A=((((x+((2ax^2 ))^(1/3) )(2a+((2a^2 x))^(1/3) )^(βˆ’1) βˆ’1)/( (x)^(1/3) βˆ’((2a))^(1/3) ))βˆ’(2a)^(βˆ’1/3) )^(βˆ’6) , (a,b)∈R^2 a- A=((16a^4 )/x^2 ) b- A=(8/(ax)) c-A=(((2a))^(1/3) /(3x^3 ))

$$\mathrm{A}=\left(\frac{\left(\mathrm{x}+\sqrt[{\mathrm{3}}]{\mathrm{2ax}^{\mathrm{2}} }\right)\left(\mathrm{2a}+\sqrt[{\mathrm{3}}]{\mathrm{2a}^{\mathrm{2}} \mathrm{x}}\right)^{βˆ’\mathrm{1}} βˆ’\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{x}}βˆ’\sqrt[{\mathrm{3}}]{\mathrm{2a}}}βˆ’\left(\mathrm{2a}\right)^{βˆ’\mathrm{1}/\mathrm{3}} \right)^{βˆ’\mathrm{6}} ,\:\left(\mathrm{a},\mathrm{b}\right)\in\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{a}-\:\mathrm{A}=\frac{\mathrm{16a}^{\mathrm{4}} }{\mathrm{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{b}-\:\mathrm{A}=\frac{\mathrm{8}}{\mathrm{ax}}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{c}-\mathrm{A}=\frac{\sqrt[{\mathrm{3}}]{\mathrm{2a}}}{\mathrm{3x}^{\mathrm{3}} } \\ $$

Question Number 146761    Answers: 1   Comments: 1

Solve the partial defferintial equation u_t =a^2 u_(xx) ,0<x<L ,t>0 u(0,t)=0 and u(L,t)=0 and u_x (x,0)=f(x)

$${Solve}\:{the}\:{partial}\:{defferintial}\:{equation} \\ $$$${u}_{{t}} ={a}^{\mathrm{2}} {u}_{{xx}} \:\:\:,\mathrm{0}<{x}<{L}\:,{t}>\mathrm{0} \\ $$$$ \\ $$$${u}\left(\mathrm{0},{t}\right)=\mathrm{0}\:\:{and}\:{u}\left({L},{t}\right)=\mathrm{0}\:\:{and}\:{u}_{{x}} \left({x},\mathrm{0}\right)={f}\left({x}\right) \\ $$

Question Number 146758    Answers: 2   Comments: 0

find forier series to half rang of f(x)=sinx ,0<x<Ο€ and prove that Ξ£_(n=1) ^∞ (1/(4n^2 βˆ’1))=(1/2)

$${find}\:{forier}\:{series}\:{to}\:{half}\:{rang}\:{of}\: \\ $$$${f}\left({x}\right)={sinx}\:\:,\mathrm{0}<{x}<\pi\:{and}\:{prove}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{4}{n}^{\mathrm{2}} βˆ’\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 146756    Answers: 2   Comments: 0

∫_( 0) ^( 1) t^2 + 1 dt

$$ \\ $$$$ \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{t}^{\mathrm{2}} \:+\:\mathrm{1}\:{dt} \\ $$

Question Number 146755    Answers: 0   Comments: 0

lim_(pβ†’+∞) Ξ£_(k=1) ^(pβˆ’1) (2/(k^2 (pβˆ’k)^2 ))=...?

$$\underset{{p}\rightarrow+\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{p}βˆ’\mathrm{1}} {\sum}}\frac{\mathrm{2}}{{k}^{\mathrm{2}} \left({p}βˆ’{k}\right)^{\mathrm{2}} }=...? \\ $$

Question Number 146752    Answers: 1   Comments: 0

∫_( 0) ^( 1) t^2 + (1/2)t βˆ’6dx

$$ \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{t}^{\mathrm{2}} +\:\frac{\mathrm{1}}{\mathrm{2}}{t}\:βˆ’\mathrm{6}{dx}\:\: \\ $$

Question Number 146746    Answers: 4   Comments: 0

∫_( 0) ^( 4) (√(16 - x^2 )) dx = ?

$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{4}} {\int}}\:\sqrt{\mathrm{16}\:-\:{x}^{\mathrm{2}} }\:{dx}\:=\:? \\ $$

Question Number 146744    Answers: 1   Comments: 0

if ∫_a ^b f(x)dx = 7 and ∫_a ^b 4 g(x)dx = βˆ’6 find ∫_a ^b (3 f(x)βˆ’8 g(x)) dx = ?

$${if}\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}{f}\left({x}\right){dx}\:=\:\mathrm{7}\:\:\:{and}\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}\mathrm{4}\:{g}\left({x}\right){dx}\:=\:βˆ’\mathrm{6} \\ $$$${find}\:\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}\left(\mathrm{3}\:{f}\left({x}\right)βˆ’\mathrm{8}\:{g}\left({x}\right)\right)\:{dx}\:=\:? \\ $$

Question Number 146736    Answers: 1   Comments: 0

1: S:= Ξ£_(n=1) ^∞ (((βˆ’1)^( nβˆ’1) )/(n.2^( n) )) =? 2: A:= Ξ£(((βˆ’1)^( nβˆ’1) )/(n^2 . 2^( n) )) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{1}:\:\:\:\:\mathrm{S}:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(βˆ’\mathrm{1}\right)^{\:{n}βˆ’\mathrm{1}} }{{n}.\mathrm{2}^{\:{n}} }\:=? \\ $$$$\:\:\:\:\:\:\mathrm{2}:\:\:\:\:\mathrm{A}:=\:\Sigma\frac{\left(βˆ’\mathrm{1}\right)^{\:{n}βˆ’\mathrm{1}} }{{n}^{\mathrm{2}} .\:\mathrm{2}^{\:{n}} }\:=? \\ $$

Question Number 146735    Answers: 2   Comments: 0

Σ_(n=1) ^∞ ((1+(1/2)+(1/3)+...+(1/n))/((n+1)(n+2)))=?

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

Question Number 146729    Answers: 0   Comments: 0

Question Number 146725    Answers: 2   Comments: 0

cos(x) βˆ™ cos(3x) = cos(5x) βˆ™ cos(7x) β‡’ x = ?

$${cos}\left({x}\right)\:\centerdot\:{cos}\left(\mathrm{3}{x}\right)\:=\:{cos}\left(\mathrm{5}{x}\right)\:\centerdot\:{cos}\left(\mathrm{7}{x}\right) \\ $$$$\Rightarrow\:{x}\:=\:? \\ $$

Question Number 146705    Answers: 1   Comments: 0

find lim_(xβ†’0) ((sin(tan(2x)βˆ’x)+1βˆ’cos(Ο€x^2 ))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{sin}\left(\mathrm{tan}\left(\mathrm{2x}\right)βˆ’\mathrm{x}\right)+\mathrm{1}βˆ’\mathrm{cos}\left(\pi\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 146702    Answers: 1   Comments: 0

Question Number 146701    Answers: 3   Comments: 2

Question Number 146697    Answers: 0   Comments: 0

βˆ€tβ‰₯βˆ’1,F(t)=(2/Ο€)∫_0 ^(Ο€/2) (√(1+tcos^2 Ο•))dΟ• 1) Show that βˆ€tβ‰€βˆ’1 F(t)=(√(1+t))F(βˆ’(1/(1+t))) 2) show that if 0≀t_1 , 0≀F(t_2 )βˆ’F(t_1 )≀((t_2 βˆ’t_1 )/4)

$$\forall{t}\geqslantβˆ’\mathrm{1},{F}\left({t}\right)=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{1}+{tcos}^{\mathrm{2}} \varphi}{d}\varphi \\ $$$$\left.\mathrm{1}\right)\:{Show}\:{that}\:\forall{t}\leqslantβˆ’\mathrm{1}\:{F}\left({t}\right)=\sqrt{\mathrm{1}+{t}}{F}\left(βˆ’\frac{\mathrm{1}}{\mathrm{1}+{t}}\right) \\ $$$$\left.\mathrm{2}\right)\:{show}\:{that}\:{if}\:\mathrm{0}\leqslant{t}_{\mathrm{1}} \:, \\ $$$$\mathrm{0}\leqslant{F}\left({t}_{\mathrm{2}} \right)βˆ’{F}\left({t}_{\mathrm{1}} \right)\leqslant\frac{{t}_{\mathrm{2}} βˆ’{t}_{\mathrm{1}} }{\mathrm{4}} \\ $$

Question Number 146689    Answers: 1   Comments: 0

∫ ((βˆ’3x+5)/( (((3x^2 βˆ’10x)^2 ))^(1/3) )) dx = ... ? Solve it without substitution method.

$$\int\:\:\frac{βˆ’\mathrm{3}{x}+\mathrm{5}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3}{x}^{\mathrm{2}} βˆ’\mathrm{10}{x}\right)^{\mathrm{2}} }}\:\:{dx}\:\:=\:\:\:...\:\:? \\ $$$${Solve}\:\:{it}\:\:{without}\:\:{substitution}\:\:{method}. \\ $$

Question Number 146687    Answers: 0   Comments: 0

E = MC^(2 ) ∫requency = E/M

$${E}\:=\:{MC}^{\mathrm{2}\:} \:\:\:\:\:\:\:\:\:\int{requency}\:=\:{E}/{M} \\ $$

Question Number 146680    Answers: 2   Comments: 0

Compare: 100^(101) and 101^(100)

$${Compare}:\:\:\mathrm{100}^{\mathrm{101}} \:\:{and}\:\:\:\mathrm{101}^{\mathrm{100}} \\ $$

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