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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

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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 }

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

Question Number 218206 Answers: 0 Comments: 0