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Question Number 157389 Answers: 0 Comments: 0

lim_(x→∞) (((6e^x^2 −20ln ∣x∣)/(6e^x^2 −20ln ∣x∣+3)))^(3e^(2x^2 ) sin (((6k)/e^x^2 ))) =26 k=?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{6}{e}^{{x}^{\mathrm{2}} } −\mathrm{20ln}\:\mid{x}\mid}{\mathrm{6}{e}^{{x}^{\mathrm{2}} } −\mathrm{20ln}\:\mid{x}\mid+\mathrm{3}}\right)^{\mathrm{3}{e}^{\mathrm{2}{x}^{\mathrm{2}} } \:\mathrm{sin}\:\left(\frac{\mathrm{6}{k}}{{e}^{{x}^{\mathrm{2}} } }\right)} =\mathrm{26} \\ $$$$\:{k}=? \\ $$$$ \\ $$

Question Number 157388 Answers: 1 Comments: 2

If m tan (θ − 30°) = n tan (θ + 12°) cos 2θ = ?

$${If}\:\:{m}\:\mathrm{tan}\:\left(\theta\:−\:\mathrm{30}°\right)\:=\:{n}\:\mathrm{tan}\:\left(\theta\:+\:\mathrm{12}°\right) \\ $$$$\mathrm{cos}\:\mathrm{2}\theta\:\:=\:? \\ $$

Question Number 157382 Answers: 1 Comments: 0

Question Number 157379 Answers: 2 Comments: 1

Question Number 157374 Answers: 0 Comments: 3

I see there are idiot comments on this forum. a comment that tells someone not to answer a member's question. this is a provocateur. what's the point of telling people not to answer questions posted by members. If the one who ordered it doesn't want to answer, then do it. no need to tell people to follow you.

$$ \\ $$I see there are idiot comments on this forum. a comment that tells someone not to answer a member's question. this is a provocateur. what's the point of telling people not to answer questions posted by members. If the one who ordered it doesn't want to answer, then do it. no need to tell people to follow you.

Question Number 157351 Answers: 3 Comments: 1

Prove by mathematical induction Σ_(r=1) ^n (1/(r(r+1))) = (n/(n+1))

$$\mathrm{Prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{r}\left({r}+\mathrm{1}\right)}\:=\:\frac{{n}}{{n}+\mathrm{1}} \\ $$

Question Number 157343 Answers: 1 Comments: 0

Question Number 157341 Answers: 0 Comments: 0

Question Number 157332 Answers: 2 Comments: 0

Question Number 157329 Answers: 2 Comments: 0

whats the value of !5

$$\:\boldsymbol{{whats}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\: \\ $$$$\:\:\:\:!\mathrm{5} \\ $$

Question Number 157324 Answers: 1 Comments: 1

Find the general solution for the equation cos 7x − cos 4x + cos x = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{for}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{cos}\:\mathrm{7}{x}\:−\:\mathrm{cos}\:\mathrm{4}{x}\:+\:\mathrm{cos}\:{x}\:=\:\mathrm{0} \\ $$

Question Number 157323 Answers: 2 Comments: 0

Prove that: Ψ_1 ((1/8)) + Ψ_2 ((5/8)) = 32G + 4π^2 Ψ-trigamma function G-catalan constant

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\Psi_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{8}}\right)\:+\:\Psi_{\mathrm{2}} \left(\frac{\mathrm{5}}{\mathrm{8}}\right)\:=\:\mathrm{32G}\:+\:\mathrm{4}\pi^{\mathrm{2}} \\ $$$$\Psi-\mathrm{trigamma}\:\mathrm{function} \\ $$$$\mathrm{G}-\mathrm{catalan}\:\mathrm{constant} \\ $$

Question Number 157315 Answers: 1 Comments: 0

Question Number 157314 Answers: 0 Comments: 1

lim_(x→∞) ((3 ln (1+5tan (4/x)))/(x (1−cos (6/x)))) =?

$$\:\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}\:\mathrm{ln}\:\left(\mathrm{1}+\mathrm{5tan}\:\frac{\mathrm{4}}{{x}}\right)}{{x}\:\left(\mathrm{1}−\mathrm{cos}\:\frac{\mathrm{6}}{{x}}\right)}\:=? \\ $$

Question Number 157309 Answers: 3 Comments: 0

∫ (dx/( (√x)+x(√(x+1)))) =?

$$\:\:\int\:\frac{{dx}}{\:\sqrt{{x}}+{x}\sqrt{{x}+\mathrm{1}}}\:=? \\ $$

Question Number 157292 Answers: 1 Comments: 6

Question Number 157283 Answers: 1 Comments: 0

Question Number 157278 Answers: 3 Comments: 0

Question Number 157272 Answers: 2 Comments: 0

for solving equation which one we use ⇒ and =, i mean where we use ⇒ and where we use, = and where we use one of them to consider wrong.

$${for}\:{solving}\:{equation}\:{which}\:{one}\:{we}\:{use} \\ $$$$\Rightarrow\:{and}\:=,\:{i}\:{mean}\:{where}\:{we}\:{use}\:\Rightarrow\:{and}\:{where}\:{we}\:{use},\:= \\ $$$${and}\:{where}\:{we}\:{use}\:{one}\:{of}\:{them}\:{to} \\ $$$${consider}\:{wrong}. \\ $$

Question Number 157268 Answers: 1 Comments: 0

prove that ∫(x^2 /((xsin x+cos x)^2 ))dx=−((xsec x)/(xsin x+cos x))+tan x+c

$${prove}\:{that} \\ $$$$\int\frac{{x}^{\mathrm{2}} }{\left({x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)^{\mathrm{2}} }{dx}=−\frac{{x}\mathrm{sec}\:{x}}{{x}\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}+\mathrm{tan}\:{x}+{c} \\ $$

Question Number 157267 Answers: 0 Comments: 3

(1+x^2 )y′′−2xy′+2y=x

$$\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}''−\mathrm{2}{xy}'+\mathrm{2}{y}={x}\: \\ $$

Question Number 157265 Answers: 1 Comments: 0

Solve for real numbers: sin(x) + cos(x) + sec(x)∙csc(x)=2+(√2)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{sin}\left(\mathrm{x}\right)\:+\:\mathrm{cos}\left(\mathrm{x}\right)\:+\:\mathrm{sec}\left(\mathrm{x}\right)\centerdot\mathrm{csc}\left(\mathrm{x}\right)=\mathrm{2}+\sqrt{\mathrm{2}} \\ $$

Question Number 157258 Answers: 0 Comments: 2

Solve for real numbers: (1/(sin^(2k) (x))) + (1/(cos^(2k) (x))) = 8 ; k∈Z

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}\boldsymbol{\mathrm{k}}} \left(\mathrm{x}\right)}\:+\:\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{2}\boldsymbol{\mathrm{k}}} \left(\mathrm{x}\right)}\:=\:\mathrm{8}\:\:\:;\:\:\:\mathrm{k}\in\mathbb{Z} \\ $$

Question Number 157257 Answers: 1 Comments: 0

∫ ((sin^6 x+cos^5 x)/(sin^2 x cos^2 x)) dx

$$\int\:\frac{\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx} \\ $$

Question Number 157251 Answers: 1 Comments: 0

# Nice Mathematics # ...calculation ... Ω :=∫_0 ^( 1) (( tanh^( −1) ((√( x)) ))/x) dx =^? (( π^( 2) )/4) −−−−−−−−−−−−− Ω :=^((√x) = t) 2∫_0 ^( 1) (( tanh^( −1) (t ))/t) dt :=^({tanh^( −1) (t )= (1/2) ln( ((1+t)/(1−t)) ) }) ∫_0 ^( 1) ((ln( 1+t )− ln(1−t ))/t) dt : = −Li_( 2) (−1 ) + Li_( 2) (1 ) :=^( {Li_( 2) (z )= Σ_(n=1) ^∞ (( z^( n) )/n^( 2) ) }) η (2) + ζ (2) := (π^( 2) /(12)) + (π^( 2) /6) = (( π^( 2) )/( 4)) ■ m.n

$$ \\ $$$$\:\:\:\:\:#\:\mathrm{Nice}\:\mathrm{Mathematics}\:# \\ $$$$\:\:\:\:\:\:\:...{calculation}\:... \\ $$$$\:\:\:\:\:\:\:\:\Omega\::=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \:\left(\sqrt{\:{x}}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:−−−−−−−−−−−−− \\ $$$$\:\:\:\:\Omega\::\overset{\sqrt{{x}}\:=\:{t}} {=}\:\mathrm{2}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \:\left({t}\:\right)}{{t}}\:{dt} \\ $$$$\:\:\:\:\:\:\:\:\::\overset{\left\{{tanh}^{\:−\mathrm{1}} \:\left({t}\:\right)=\:\frac{\mathrm{1}}{\mathrm{2}}\:{ln}\left(\:\frac{\mathrm{1}+{t}}{\mathrm{1}−{t}}\:\right)\:\right\}} {=}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\:\mathrm{1}+{t}\:\right)−\:{ln}\left(\mathrm{1}−{t}\:\right)}{{t}}\:{dt} \\ $$$$\:\:\:\::\:\:=\:\:−\mathrm{Li}_{\:\mathrm{2}} \:\left(−\mathrm{1}\:\right)\:+\:\mathrm{Li}_{\:\mathrm{2}} \:\left(\mathrm{1}\:\right) \\ $$$$\:\:\:\:\::\overset{\:\left\{\mathrm{Li}_{\:\mathrm{2}} \:\left({z}\:\right)=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{z}^{\:{n}} }{{n}^{\:\mathrm{2}} }\:\right\}} {=}\:\:\eta\:\left(\mathrm{2}\right)\:+\:\zeta\:\left(\mathrm{2}\right)\: \\ $$$$\:\:\:\:\::=\:\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{12}}\:+\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{6}}\:\:=\:\frac{\:\pi^{\:\mathrm{2}} }{\:\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 157247 Answers: 3 Comments: 0

F(x,y)=x^2 −2xy+6y^2 −12x+2y+45 find x &y such that F(x,y) minimum

$${F}\left({x},{y}\right)={x}^{\mathrm{2}} −\mathrm{2}{xy}+\mathrm{6}{y}^{\mathrm{2}} −\mathrm{12}{x}+\mathrm{2}{y}+\mathrm{45} \\ $$$${find}\:{x}\:\&{y}\:{such}\:{that}\:{F}\left({x},{y}\right)\:{minimum} \\ $$

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