Question and Answers Forum

AllQuestion and Answers: Page 566

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

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

Question Number 157241 Answers: 0 Comments: 0