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Question Number 117286 Answers: 2 Comments: 0

A person wants to invite his 6 friends in a Dinner party. He has 3 person to send letter to them.In how many ways he can invite his 6 friends?

$${A}\:{person}\:{wants}\:{to}\:{invite}\:{his}\:\mathrm{6}\:{friends}\:{in}\:{a}\:{Dinner}\:{party}. \\ $$$${He}\:{has}\:\mathrm{3}\:{person}\:{to}\:{send}\:{letter}\:{to}\:{them}.{In}\:{how}\:{many}\:{ways} \\ $$$${he}\:{can}\:{invite}\:{his}\:\mathrm{6}\:{friends}? \\ $$

Question Number 117205 Answers: 0 Comments: 0

∫_0 ^( 2π) ((cos^2 3x)/(1−2a∙cosx+a^2 ))dx − ? (a∈C/{−1; 1}) I need a solution through complex analysis

$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \frac{\boldsymbol{{cos}}^{\mathrm{2}} \mathrm{3}\boldsymbol{{x}}}{\mathrm{1}−\mathrm{2}\boldsymbol{{a}}\centerdot\boldsymbol{{cosx}}+\boldsymbol{{a}}^{\mathrm{2}} }\boldsymbol{{dx}}\:−\:?\:\left(\boldsymbol{{a}}\in\boldsymbol{{C}}/\left\{−\mathrm{1};\:\mathrm{1}\right\}\right) \\ $$$$\boldsymbol{{I}}\:\boldsymbol{{need}}\:\boldsymbol{{a}}\:\boldsymbol{{solution}}\:\boldsymbol{{through}}\:\boldsymbol{{complex}}\:\boldsymbol{{analysis}} \\ $$

Question Number 117204 Answers: 1 Comments: 2

Question Number 117215 Answers: 0 Comments: 1

v2.220: Forum viewing improvement • After update you will see options to zoom out of post with longer lines to view whole posts without scrolling. Similar option are added while editing in ′Cursor&Page′ menu

$$\mathrm{v2}.\mathrm{220}:\:\mathrm{Forum}\:\mathrm{viewing}\:\mathrm{improvement} \\ $$$$\bullet\:\mathrm{After}\:\mathrm{update}\:\mathrm{you}\:\mathrm{will}\:\mathrm{see}\:\mathrm{options} \\ $$$$\mathrm{to}\:\mathrm{zoom}\:\mathrm{out}\:\mathrm{of}\:\mathrm{post}\:\mathrm{with}\:\mathrm{longer}\:\mathrm{lines} \\ $$$$\mathrm{to}\:\mathrm{view}\:\mathrm{whole}\:\mathrm{posts}\:\mathrm{without}\:\mathrm{scrolling}. \\ $$$$\mathrm{Similar}\:\mathrm{option}\:\mathrm{are}\:\mathrm{added}\:\mathrm{while} \\ $$$$\mathrm{editing}\:\mathrm{in}\:'\mathrm{Cursor\&Page}'\:\mathrm{menu} \\ $$

Question Number 117192 Answers: 1 Comments: 0

Assuming you have enough coins of 1,5,10,25,and 50cents. In how many ways can you make a change for 1dollar.

$$\mathrm{Assuming}\:\mathrm{you}\:\mathrm{have}\:\mathrm{enough}\:\mathrm{coins}\: \\ $$$$\mathrm{of}\:\mathrm{1},\mathrm{5},\mathrm{10},\mathrm{25},\mathrm{and}\:\mathrm{50cents}.\:\mathrm{In}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{you}\:\mathrm{make}\:\mathrm{a}\:\mathrm{change}\:\mathrm{for}\:\mathrm{1dollar}. \\ $$

Question Number 117191 Answers: 0 Comments: 0

∫sin^(3/2) (x)dx

$$\int\mathrm{sin}^{\frac{\mathrm{3}}{\mathrm{2}}} \left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 117188 Answers: 2 Comments: 1

Given r+(1/r) = (√2) , then r^8 +(1/r^8 ) = ?

$${Given}\:{r}+\frac{\mathrm{1}}{{r}}\:=\:\sqrt{\mathrm{2}}\:,\:{then}\:{r}^{\mathrm{8}} +\frac{\mathrm{1}}{{r}^{\mathrm{8}} }\:=\:?\: \\ $$

Question Number 117176 Answers: 4 Comments: 1

∫_(π/4) ^(π/2) ((4cot x+1 )/(4−cot x)) dx =?

$$\underset{\pi/\mathrm{4}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{4cot}\:{x}+\mathrm{1}\:}{\mathrm{4}−\mathrm{cot}\:{x}}\:{dx}\:=? \\ $$

Question Number 117172 Answers: 1 Comments: 0

Question Number 117171 Answers: 1 Comments: 0

lim_(x→0) (((1+x tan^(−1) (x))/(cosh (x))))^(1/(tan^2 (x))) =?

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

Question Number 117164 Answers: 2 Comments: 0

lim_(n→∞) (((n+9)/(2n−1)))^n =?

$$\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{n}+\mathrm{9}}{\mathrm{2n}−\mathrm{1}}\right)^{\mathrm{n}} =? \\ $$

Question Number 117163 Answers: 2 Comments: 0

∫ ((4 dx)/(x (√(x^2 −1)))) =?

$$\:\:\:\int\:\frac{\mathrm{4}\:\mathrm{dx}}{\mathrm{x}\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\:=? \\ $$

Question Number 117144 Answers: 1 Comments: 0

S_n =Π_(k=1) ^n cos(kx)=?????? please help

$$\boldsymbol{{S}}_{\boldsymbol{{n}}} =\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\prod}}\boldsymbol{{cos}}\left(\boldsymbol{{kx}}\right)=?????? \\ $$$$\boldsymbol{{please}}\:\boldsymbol{{help}} \\ $$

Question Number 117143 Answers: 1 Comments: 0

Question Number 117142 Answers: 0 Comments: 0

Question Number 117140 Answers: 1 Comments: 0

Question Number 117216 Answers: 1 Comments: 0

∫_0 ^(π/2) (dx/( (√(sin x)))) =?

$$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{\mathrm{dx}}{\:\sqrt{\mathrm{sin}\:\mathrm{x}}}\:=?\: \\ $$$$ \\ $$

Question Number 117213 Answers: 3 Comments: 0

∫_0 ^( ∞) ((tan^(−1) ((x/4))−tan^(−1) ((x/6)))/x) dx =?

$$\:\:\underset{\mathrm{0}} {\overset{\:\:\:\:\:\:\:\infty} {\int}}\:\frac{\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{4}}\right)−\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{6}}\right)}{\mathrm{x}}\:\mathrm{dx}\:=? \\ $$

Question Number 117184 Answers: 0 Comments: 0

let I be an interval.prove that c^((∞)) (I)=∩_(n=1) ^∞ c^n (I)

$${let}\:{I}\:{be}\:{an}\:{interval}.{prove}\:{that} \\ $$$${c}^{\left(\infty\right)} \left({I}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\cap}}{c}^{{n}} \left({I}\right) \\ $$

Question Number 117122 Answers: 0 Comments: 4

Given a,b,c ∈R^3 such that abc=1. Show that: (a−1+(1/b))(b−1+(1/c))(c−1+(1/a))≤1

$$\mathrm{Given}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\in\mathbb{R}^{\mathrm{3}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{abc}=\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\left(\mathrm{a}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{b}}\right)\left(\mathrm{b}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{c}}\right)\left(\mathrm{c}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{a}}\right)\leqslant\mathrm{1} \\ $$

Question Number 117116 Answers: 0 Comments: 2

x , y ,z , t ∈ Z. x and y are x are respectively the divisor of y and t. Justify the existence of k ∈ Z such that yt=xzk. Deduct that x^(m ) is a divisor of y^m where m ∈ N.

$${x}\:,\:{y}\:,{z}\:,\:{t}\:\in\:\mathbb{Z}. \\ $$$${x}\:{and}\:{y}\:{are}\:{x}\:{are}\:{respectively}\:{the} \\ $$$${divisor}\:{of}\:{y}\:{and}\:{t}. \\ $$$${Justify}\:{the}\:{existence}\:{of}\:{k}\:\in\:\mathbb{Z}\:{such} \\ $$$${that}\:{yt}={xzk}. \\ $$$${Deduct}\:{that}\:{x}^{{m}\:} {is}\:{a}\:{divisor}\:{of}\:{y}^{{m}} \\ $$$${where}\:{m}\:\in\:\mathbb{N}. \\ $$

Question Number 117107 Answers: 1 Comments: 0

Question Number 117101 Answers: 2 Comments: 0

If sin^2 θ and cos^2 θ are the roots of quadratic equation, find the equation.

$$\mathrm{If}\:\mathrm{sin}^{\mathrm{2}} \theta\:\mathrm{and}\:\mathrm{cos}^{\mathrm{2}} \theta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\: \\ $$$$\mathrm{of}\:\mathrm{quadratic}\:\mathrm{equation},\:\mathrm{find}\:\mathrm{the}\:\mathrm{equation}. \\ $$

Question Number 117100 Answers: 3 Comments: 0

∫ sin^6 (2x) cos^6 (2x) dx =?

$$\int\:\mathrm{sin}\:^{\mathrm{6}} \left(\mathrm{2x}\right)\:\mathrm{cos}\:^{\mathrm{6}} \left(\mathrm{2x}\right)\:\mathrm{dx}\:=? \\ $$

Question Number 117148 Answers: 2 Comments: 0

∫ x^6 e^(−4x^2 ) dx =?

$$\:\:\:\int\:\mathrm{x}^{\mathrm{6}} \:\mathrm{e}^{−\mathrm{4x}^{\mathrm{2}} } \:\mathrm{dx}\:=? \\ $$

Question Number 117147 Answers: 2 Comments: 0

lim_(x→0) (((cosh (2x))/(cosh (x))))^(1/x^2 ) =?

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

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