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Question Number 216613 Answers: 0 Comments: 1

∫_a ^( x) ((1−bln (x/a))/( (√(1−(1−bln (x/a))^2 )))) dx

$$\int_{{a}} ^{\:{x}} \frac{\mathrm{1}−{b}\mathrm{ln}\:\frac{{x}}{{a}}}{\:\sqrt{\mathrm{1}−\left(\mathrm{1}−{b}\mathrm{ln}\:\frac{{x}}{{a}}\right)^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 216607 Answers: 1 Comments: 0

Question Number 216596 Answers: 1 Comments: 0

∫∫_(D) ((sin(x^2 +y^2 )+tan(x^2 +y^2 ))/(cos(x^2 +y^2 )+tan(x^2 +y^2 )))dxdy D=[0,(π/4)]×[0,(π/4)]

$$\underset{\mathcal{D}} {\int\int}\:\:\:\frac{\mathrm{sin}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{\mathrm{cos}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}\mathrm{d}{x}\mathrm{d}{y} \\ $$$$\mathcal{D}=\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right]×\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$

Question Number 216593 Answers: 2 Comments: 0

Question Number 216592 Answers: 0 Comments: 0

if we have the following system: ((tanx)/(tany))=a x±y=α we have the general candition: −1≤(((a−1)/(a+1)))sinα≤1 if you apply the general candition by the following system it does not give us the reality, despite this system have the solution. ((tanx)/(tany))=−3 x−y=(π/2)

$${if}\:{we}\:{have}\:{the}\:{following}\:{system}: \\ $$$$\frac{{tanx}}{{tany}}={a} \\ $$$${x}\pm{y}=\alpha \\ $$$${we}\:{have}\:{the}\:{general}\:{candition}: \\ $$$$−\mathrm{1}\leqslant\left(\frac{{a}−\mathrm{1}}{{a}+\mathrm{1}}\right){sin}\alpha\leqslant\mathrm{1} \\ $$$${if}\:{you}\:{apply}\:{the}\:{general}\:{candition}\:{by} \\ $$$${the}\:{following}\:{system}\:{it}\:{does}\:{not}\:{give} \\ $$$${us}\:{the}\:{reality},\:{despite}\:{this}\:{system}\:{have} \\ $$$${the}\:{solution}. \\ $$$$\frac{{tanx}}{{tany}}=−\mathrm{3} \\ $$$${x}−{y}=\frac{\pi}{\mathrm{2}} \\ $$

Question Number 216587 Answers: 1 Comments: 4

Question Number 216585 Answers: 0 Comments: 3

is this right? i had let θ= { ((tan^(−1) ((d/c))),((c>0,d>0))),((π−tan^(−1) ((d/c))),((c<0,d>0))),((−π+tan^(−1) ((d/c))),((c<0,d<0))),((−tan^(−1) ((d/c))),((c>0,d<0))) :} before i calculated below (a+bi)^(c+di) =∣a+di∣^(c+di) e^(i(c+di)θ) =∣a+bi∣^c ∣a+bi∣^di e^(icθ) e^(−dθ) =((√(a^2 +b^2 )))^c ((√(a^2 +b^2 )))^di e^(icθ) e^(−dθ) =((√(a^2 +b^2 )))^c e^((1/2)idln(a^2 +b^2 )) e^(icθ) e^(−dθ) =((√(a^2 +b^2 )))^c e^(−dθ) e^(i(dln(a^2 +b^2 )+cθ)) =((√(a^2 +b^2 )))^c e^(−dθ) (cos(dln(a^2 +b^2 )+cθ)+isin(dln(a^2 +b^2 )+cθ))

Question Number 216582 Answers: 0 Comments: 3

using first principle solve y=((x+2)/( (√x)+2)) is it possible with first principle

$$ \\ $$$$\mathrm{using}\:\mathrm{first}\:\mathrm{principle}\:\mathrm{solve} \\ $$$$\mathrm{y}=\frac{\mathrm{x}+\mathrm{2}}{\:\sqrt{\mathrm{x}}+\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{with}\:\mathrm{first}\:\mathrm{principle} \\ $$

Question Number 216579 Answers: 0 Comments: 0

∫_0 ^(π/4) ∫_0 ^(π/4) ((tan(x^2 +y^2 )+sin(x^2 +y^2 ))/(tan(x^2 +y^2 )+cos(x^2 +y^2 )))dxdy

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{sin}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{\mathrm{tan}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)+\mathrm{cos}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{dxdy} \\ $$

Question Number 216560 Answers: 1 Comments: 0

An object of mass M, initially at rest at the coordinate origin, explodes into three parts. Fragment A has a mass M/2, and fragments B and C have a mass M/4 each. After the explosion, fragment A moves in the +X direction at 10 m/s and fragment B moves in the +Y direction at 8 m/s. Find the direction and speed of fragment C

Question Number 216542 Answers: 0 Comments: 1

if y=cosu then prove that y′=sinu∙u′ by newton′s formula.

$${if}\:{y}={cosu}\:{then}\:{prove}\:{that}\:{y}'={sinu}\centerdot{u}' \\ $$$${by}\:{newton}'{s}\:{formula}. \\ $$

Question Number 216538 Answers: 0 Comments: 4

Find all integer solutions of 3^m =2n^2 +1. I only found m=1, 2, 5 by computer from m=1 to m=30000. Is there any greater solutions?

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\mathrm{3}^{{m}} =\mathrm{2}{n}^{\mathrm{2}} +\mathrm{1}. \\ $$$$ \\ $$$${I}\:{only}\:{found}\:{m}=\mathrm{1},\:\mathrm{2},\:\mathrm{5}\:{by}\:{computer} \\ $$$${from}\:{m}=\mathrm{1}\:{to}\:{m}=\mathrm{30000}. \\ $$$${Is}\:{there}\:{any}\:{greater}\:{solutions}? \\ $$

Question Number 216537 Answers: 0 Comments: 0

using first principle solve y=((x+2)/( (√x)+2)) is it possible with first principle

$$\mathrm{using}\:\mathrm{first}\:\mathrm{principle}\:\mathrm{solve} \\ $$$$\mathrm{y}=\frac{\mathrm{x}+\mathrm{2}}{\:\sqrt{\mathrm{x}}+\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{with}\:\mathrm{first}\:\mathrm{principle} \\ $$

Question Number 216534 Answers: 0 Comments: 24

Question Number 216572 Answers: 1 Comments: 0

Question Number 216532 Answers: 2 Comments: 0

Let f :R_+ →R such as f(xy)=f(x)+f(y) 1) Prove that f is derivable iff f is derivable at x=1. 2) Prove that if so, f(x)=Log_a x) where a is positive value to precise

$${Let}\:{f}\::\mathbb{R}_{+} \rightarrow\mathbb{R}\:{such}\:{as}\:{f}\left({xy}\right)={f}\left({x}\right)+{f}\left({y}\right) \\ $$$$\left.\mathrm{1}\right)\:{Prove}\:{that}\:\:{f}\:{is}\:{derivable}\:{iff}\:\: \\ $$$${f}\:{is}\:{derivable}\:{at}\:{x}=\mathrm{1}. \\ $$$$\left.\mathrm{2}\left.\right)\:{Prove}\:{that}\:{if}\:{so},\:{f}\left({x}\right)={Log}_{{a}} {x}\right)\: \\ $$$${where}\:{a}\:{is}\:{positive}\:{value}\:{to}\:{precise} \\ $$

Question Number 216526 Answers: 1 Comments: 0

Question Number 216525 Answers: 2 Comments: 0

Question Number 216515 Answers: 2 Comments: 0

Question Number 216513 Answers: 0 Comments: 0

Prove or disprove that: If p=(√(Σ_(k=0) ^n 3^k )) (n>0) is an integer, then p is prime.

$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{that}: \\ $$$$\mathrm{If}\:{p}=\sqrt{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\mathrm{3}^{{k}} }\:\left({n}>\mathrm{0}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer},\:\mathrm{then}\:{p}\:\mathrm{is}\:\mathrm{prime}. \\ $$

Question Number 216507 Answers: 0 Comments: 1

Question Number 216493 Answers: 1 Comments: 0

Res_(z=c) {f(z)}=(1/(2πi)) ∮_( C) f(z)dz Res_(z=1) {((z^(21) +z^2 +z+1)/((z−1)^3 ))}=(1/(2πi)) ∮_( C) ((z^(21) +z^2 +z+1)/((z−1)^3 ))dz (1/(2πi)) ∮_( C) (((z^(21) +z^2 +z+1)/((z−1)^2 ))/(z−1))dz=lim_(z→1) ((z^(21) +z^2 +z+1)/((z−1)^2 )) L′hosiptal :) lim_(z→1) ((21z^(20) +2z+1)/(2(z−1))) and... Twice!! lim_(z→1) ((420z^(19) +2)/2)=211 ∴Res_(z=1) {f(z)}=211 ★Caution★ f(α)′′=′′(1/(2πi)) ∮_( C) ((f(z))/(z−α)) dz Why did I use big quotes for this equation?? because the conditions for establshing this equation are that path C must be a simple closed curve and there must be no singularity in path C

$$\mathrm{Res}_{{z}={c}} \left\{{f}\left({z}\right)\right\}=\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:\mathrm{C}} \:{f}\left({z}\right)\mathrm{d}{z} \\ $$$$\mathrm{Res}_{{z}=\mathrm{1}} \left\{\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{3}} }\right\}=\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:{C}} \:\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{3}} }\mathrm{d}{z} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:{C}} \:\:\frac{\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{2}} }}{{z}−\mathrm{1}}\mathrm{d}{z}=\underset{{z}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\:\frac{{z}^{\mathrm{21}} +{z}^{\mathrm{2}} +{z}+\mathrm{1}}{\left({z}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{L}'\mathrm{hosiptal}\::\right) \\ $$$$\underset{{z}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{21}{z}^{\mathrm{20}} +\mathrm{2}{z}+\mathrm{1}}{\mathrm{2}\left({z}−\mathrm{1}\right)}\:\:\mathrm{and}...\:\mathrm{Twice}!! \\ $$$$\underset{{z}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{420}{z}^{\mathrm{19}} +\mathrm{2}}{\mathrm{2}}=\mathrm{211} \\ $$$$\therefore\mathrm{Res}_{{z}=\mathrm{1}} \left\{{f}\left({z}\right)\right\}=\mathrm{211} \\ $$$$\bigstar\mathrm{Caution}\bigstar \\ $$$${f}\left(\alpha\right)''=''\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\oint_{\:{C}} \:\:\frac{{f}\left({z}\right)}{{z}−\alpha}\:\mathrm{d}{z}\: \\ $$$$\mathrm{Why}\:\mathrm{did}\:\mathrm{I}\:\mathrm{use}\:\mathrm{big}\:\mathrm{quotes}\:\mathrm{for}\:\mathrm{this}\: \\ $$$$\mathrm{equation}?? \\ $$$$\mathrm{because}\:\mathrm{the}\:\mathrm{conditions}\:\mathrm{for}\:\mathrm{establshing} \\ $$$$\mathrm{this}\:\mathrm{equation}\:\mathrm{are}\:\mathrm{that}\:\mathrm{path}\:{C}\: \\ $$$$\mathrm{must}\:\mathrm{be}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{curve} \\ $$$$\mathrm{and}\:\mathrm{there}\:\mathrm{must}\:\mathrm{be}\:\mathrm{no}\:\mathrm{singularity} \\ $$$$\mathrm{in}\:\mathrm{path}\:\mathrm{C} \\ $$

Question Number 216491 Answers: 2 Comments: 0

Question Number 216489 Answers: 1 Comments: 1

find residuo ((x^(21) +x^2 +x+1)/((x−1)^3 ))

$${find}\:\:{residuo} \\ $$$$\:\:\:\:\:\:\:\:\frac{{x}^{\mathrm{21}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 216487 Answers: 1 Comments: 2

Find the value of ω^7 + ω^8 + ω^(12) where ω is omega function.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\omega^{\mathrm{7}} \:\:+\:\:\omega^{\mathrm{8}} \:\:+\:\:\omega^{\mathrm{12}} \:\:\mathrm{where} \\ $$$$\omega\:\:\mathrm{is}\:\mathrm{omega}\:\mathrm{function}. \\ $$

Question Number 216486 Answers: 1 Comments: 0

∫_( 0) ^( 1) x(√(x ((x ((x ((x ...))^(1/5) ))^(1/4) ))^(1/3) )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{x}\sqrt{\mathrm{x}\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:\sqrt[{\mathrm{4}}]{\mathrm{x}\:\:\sqrt[{\mathrm{5}}]{\mathrm{x}\:...}}}}\:\:\mathrm{dx} \\ $$

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