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AllQuestion and Answers: Page 62

Question Number 218356 Answers: 2 Comments: 0

Question Number 218357 Answers: 3 Comments: 0

Question Number 218345 Answers: 1 Comments: 0

Question Number 218344 Answers: 0 Comments: 1

Question Number 218349 Answers: 3 Comments: 0

Question Number 218354 Answers: 0 Comments: 0

Question Number 218331 Answers: 1 Comments: 0

∫_0 ^∞ ((x^2 sin (x))/(1+x^4 )) dx = ?

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} \:\mathrm{sin}\:\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:=\:? \\ $$

Question Number 218328 Answers: 1 Comments: 0

Question Number 218310 Answers: 2 Comments: 1

10 couples are invited to a dinner and should be seated at a round table. in how many ways can the host do this, 1) generally. 2) if the husband and the wife of each couple should sit together. 3) if no two men should sit next to each other. 4) as 3), but two speicial couples should not sit next to each other. it means that the husband from couple 1 may not sit next to the wife from couple 2 and the husband from couple 2 may not sit next to the wife from couple 1. but certainly the husbands of both couples may sit next to their own wifes.

$$\mathrm{10}\:{couples}\:{are}\:{invited}\:{to}\:{a}\:{dinner} \\ $$$${and}\:{should}\:{be}\:{seated}\:{at}\:{a}\:{round}\:{table}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{host}\:{do} \\ $$$${this}, \\ $$$$\left.\mathrm{1}\right)\:{generally}. \\ $$$$\left.\mathrm{2}\right)\:{if}\:{the}\:{husband}\:{and}\:{the}\:{wife}\:{of} \\ $$$$\:\:\:\:\:{each}\:{couple}\:{should}\:{sit}\:{together}. \\ $$$$\left.\mathrm{3}\right)\:{if}\:{no}\:{two}\:{men}\:{should}\:{sit}\:{next} \\ $$$$\:\:\:\:\:{to}\:{each}\:{other}. \\ $$$$\left.\mathrm{4}\left.\right)\:{as}\:\mathrm{3}\right),\:{but}\:{two}\:{speicial}\:{couples}\: \\ $$$$\:\:\:\:\:{should}\:{not}\:{sit}\:{next}\:{to}\:{each}\:{other}. \\ $$$$\:\:\:\:\underline{\:{it}\:{means}\:}{that}\:{the}\:{husband}\:{from}\: \\ $$$$\:\:\:\:\:{couple}\:\mathrm{1}\:{may}\:{not}\:{sit}\:{next}\:{to}\:{the} \\ $$$$\:\:\:\:\:{wife}\:{from}\:{couple}\:\mathrm{2}\:{and}\:{the} \\ $$$$\:\:\:\:\:\:{husband}\:{from}\:{couple}\:\mathrm{2}\:{may}\:{not} \\ $$$$\:\:\:\:\:\:{sit}\:{next}\:{to}\:{the}\:{wife}\:{from}\:{couple}\:\mathrm{1}. \\ $$$$\:\:\:\:\:\:{but}\:{certainly}\:{the}\:{husbands}\:{of} \\ $$$$\:\:\:\:\:\:{both}\:{couples}\:{may}\:{sit}\:{next}\:{to}\:{their} \\ $$$$\:\:\:\:\:\:{own}\:{wifes}. \\ $$

Question Number 218322 Answers: 1 Comments: 0

Evaluate ∫_0 ^(π/2) ((sin(x))/(sin^3 (x)+cos^3 (x))) dx.

$$\mathrm{Evaluate}\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\left({x}\right)}{\mathrm{sin}^{\mathrm{3}} \left({x}\right)+\mathrm{cos}^{\mathrm{3}} \left({x}\right)}\:{dx}. \\ $$

Question Number 218318 Answers: 1 Comments: 3

Question Number 218317 Answers: 1 Comments: 0

Question Number 218280 Answers: 1 Comments: 0

Question Number 218279 Answers: 2 Comments: 0

Evaluate: (4^(log_(5/4) 4) /5^(log_(5/4) 5) ) Show workings please.

$$\mathrm{Evaluate}: \\ $$$$\:\:\:\:\:\frac{\mathrm{4}^{\mathrm{log}_{\frac{\mathrm{5}}{\mathrm{4}}} \mathrm{4}} }{\mathrm{5}^{\mathrm{log}_{\frac{\mathrm{5}}{\mathrm{4}}} \mathrm{5}} } \\ $$$$\mathrm{Show}\:\mathrm{workings}\:\mathrm{please}. \\ $$

Question Number 218278 Answers: 0 Comments: 0

Question Number 218312 Answers: 1 Comments: 0

Question Number 218311 Answers: 1 Comments: 0

Question Number 218267 Answers: 1 Comments: 1

Question Number 218265 Answers: 1 Comments: 0

can interpret the metric Tensor g_(μν) is kinda distance function at curved Surface ?? ex. Euclidean space g_(μν) = ((1,0,0),(0,1,0),(0,0,1) ) Sphere g_(μν) = ((( 1),( 0),( 0)),(( 0),( r^2 ),( 0)),(( 0),( 0),(r^2 sin^2 (θ))) )

$$\mathrm{can}\:\mathrm{interpret}\:\mathrm{the}\:\mathrm{metric}\:\mathrm{Tensor}\:\boldsymbol{\mathrm{g}}_{\mu\nu} \:\mathrm{is}\: \\ $$$$\mathrm{kinda}\:\mathrm{distance}\:\mathrm{function}\:\mathrm{at}\:\mathrm{curved}\:\mathrm{Surface}\:?? \\ $$$$\mathrm{ex}.\:\mathrm{Euclidean}\:\mathrm{space}\:\boldsymbol{\mathrm{g}}_{\mu\nu} =\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}\end{pmatrix} \\ $$$$\mathrm{Sphere}\:\boldsymbol{\mathrm{g}}_{\mu\nu} =\begin{pmatrix}{\:\mathrm{1}}&{\:\:\:\:\mathrm{0}}&{\:\:\:\:\:\:\:\mathrm{0}}\\{\:\mathrm{0}}&{\:\:\:\:{r}^{\mathrm{2}} }&{\:\:\:\:\:\:\:\mathrm{0}}\\{\:\mathrm{0}}&{\:\:\:\:\:\mathrm{0}}&{{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\end{pmatrix} \\ $$

Question Number 218262 Answers: 1 Comments: 1

Question Number 218257 Answers: 1 Comments: 2

Question Number 218256 Answers: 2 Comments: 0

Question Number 218255 Answers: 0 Comments: 0

Question Number 218236 Answers: 1 Comments: 3

Question Number 218221 Answers: 1 Comments: 7

Question Number 218208 Answers: 1 Comments: 1

This question is really important Prove or disprove that lim_(n→∞) ((3^n m+3^(n−1) )/2^(⌈(n/2)⌉) ) + (3^(n−1) /2^n ) the limit exists for m ∈ N \B where B = {n ∣ log_2 (n) ∈ N }

$${This}\:{question}\:{is}\:{really}\:{important} \\ $$$${Prove}\:{or}\:{disprove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}^{{n}} {m}+\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{\lceil\frac{{n}}{\mathrm{2}}\rceil} }\:+\:\frac{\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{{n}} }\: \\ $$$$\:{the}\:{limit}\:{exists}\:{for}\:{m}\:\in\:{N}\:\backslash{B} \\ $$$${where}\:{B}\:=\:\left\{{n}\:\mid\:{log}_{\mathrm{2}} \left({n}\right)\:\in\:{N}\:\right\} \\ $$

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