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Question Number 207451 Answers: 0 Comments: 0

find the volume in the first quadrant of the solid obtained by rotating the region bounded by the curves x = sinh(y) , x = cosh(y) about y axis (use washer method) ?

$${find}\:{the}\:{volume}\:{in}\:{the}\:{first}\:{quadrant} \\ $$$$\:{of}\:{the}\:{solid}\:{obtained}\:{by}\:{rotating} \\ $$$${the}\:{region}\:{bounded}\:{by}\:{the}\:{curves}\: \\ $$$${x}\:=\:{sinh}\left({y}\right)\:,\:{x}\:=\:{cosh}\left({y}\right)\:{about}\:{y}\:{axis}\:\left({use}\:{washer}\:{method}\right)\:? \\ $$

Question Number 207442 Answers: 1 Comments: 3

If y=f(x), (d^2 x/dy^2 )=e^(y+1) , and the tangent line to the curve of the function f(x) on the point (x_1 ,−1) is paralel to the straight line g(x)=x−3, then find f′(x).

$$\mathrm{If}\:{y}={f}\left({x}\right),\:\frac{{d}^{\mathrm{2}} {x}}{{dy}^{\mathrm{2}} }={e}^{{y}+\mathrm{1}} ,\:\mathrm{and}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{on}\:\mathrm{the}\:\mathrm{point} \\ $$$$\left({x}_{\mathrm{1}} ,−\mathrm{1}\right)\:\mathrm{is}\:\mathrm{paralel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{straight}\:\mathrm{line}\:{g}\left({x}\right)={x}−\mathrm{3},\:\mathrm{then}\:\mathrm{find}\:{f}'\left({x}\right). \\ $$

Question Number 207436 Answers: 1 Comments: 2

a, b∈N_+ , ((b+1)/a)+((a+1)/b)∈Z. Prove that (a, b)≤(√(a+b.))

$${a},\:{b}\in\mathbb{N}_{+} ,\:\frac{{b}+\mathrm{1}}{{a}}+\frac{{a}+\mathrm{1}}{{b}}\in\mathbb{Z}.\:\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\left({a},\:{b}\right)\leqslant\sqrt{{a}+{b}.} \\ $$

Question Number 207434 Answers: 0 Comments: 0

Relating to question 207407 x^3 −12x^2 +27x−17=0 Let x=t+4 t^3 −21t−37=0 The Trigonometric Solution gives these: x_1 =4−2(√7)cos ((π+2sin^(−1) ((37(√7))/(98)))/6) x_2 =4−2(√7)sin ((sin^(−1) ((37(√7))/(98)))/3) x_3 =4+2(√7)sin ((π+sin^(−1) ((37(√7))/(98)))/3) Prove these identities: x_1 =2−((1+2sin (π/(18)))/(2cos (π/9))) x_2 =2+((1+2cos (π/9))/(2cos ((2π)/9))) x_3 =((1+2(√3)sin ((2π)/9))/(2sin (π/(18))))

$$\mathrm{Relating}\:\mathrm{to}\:\mathrm{question}\:\mathrm{207407} \\ $$$${x}^{\mathrm{3}} −\mathrm{12}{x}^{\mathrm{2}} +\mathrm{27}{x}−\mathrm{17}=\mathrm{0} \\ $$$$\mathrm{Let}\:{x}={t}+\mathrm{4} \\ $$$${t}^{\mathrm{3}} −\mathrm{21}{t}−\mathrm{37}=\mathrm{0} \\ $$$$\mathrm{The}\:\mathrm{Trigonometric}\:\mathrm{Solution}\:\mathrm{gives}\:\mathrm{these}: \\ $$$${x}_{\mathrm{1}} =\mathrm{4}−\mathrm{2}\sqrt{\mathrm{7}}\mathrm{cos}\:\frac{\pi+\mathrm{2sin}^{−\mathrm{1}} \:\frac{\mathrm{37}\sqrt{\mathrm{7}}}{\mathrm{98}}}{\mathrm{6}} \\ $$$${x}_{\mathrm{2}} =\mathrm{4}−\mathrm{2}\sqrt{\mathrm{7}}\mathrm{sin}\:\frac{\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{37}\sqrt{\mathrm{7}}}{\mathrm{98}}}{\mathrm{3}} \\ $$$${x}_{\mathrm{3}} =\mathrm{4}+\mathrm{2}\sqrt{\mathrm{7}}\mathrm{sin}\:\frac{\pi+\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{37}\sqrt{\mathrm{7}}}{\mathrm{98}}}{\mathrm{3}} \\ $$$$\mathrm{Prove}\:\mathrm{these}\:\mathrm{identities}: \\ $$$${x}_{\mathrm{1}} =\mathrm{2}−\frac{\mathrm{1}+\mathrm{2sin}\:\frac{\pi}{\mathrm{18}}}{\mathrm{2cos}\:\:\frac{\pi}{\mathrm{9}}} \\ $$$${x}_{\mathrm{2}} =\mathrm{2}+\frac{\mathrm{1}+\mathrm{2cos}\:\frac{\pi}{\mathrm{9}}}{\mathrm{2cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}} \\ $$$${x}_{\mathrm{3}} =\frac{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{3}}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}{\mathrm{2sin}\:\frac{\pi}{\mathrm{18}}} \\ $$

Question Number 207435 Answers: 1 Comments: 0

y=e^t −e^(−t) and x = e^t +e^(−t) does this parametric equation resembles circle or ellipse or hyperbola or parabola and why?

$${y}={e}^{{t}} −{e}^{−{t}} \:{and}\:{x}\:=\:{e}^{{t}} +{e}^{−{t}} \:{does}\:{this}\:{parametric}\:{equation}\:{resembles} \\ $$$$\:{circle}\:{or}\:{ellipse}\:{or}\:{hyperbola}\:{or}\:{parabola}\:{and}\:{why}? \\ $$

Question Number 207426 Answers: 1 Comments: 0

Question Number 207424 Answers: 0 Comments: 0

f_n (x):=∫e^((2x)/3) ((cos(x))/( (cos(x)+sin(x))^(n/3) ))dx=...? for n=1, i found f_1 (x)=(3/4)e^((2x)/3) (cos(x)+sin(x))^(2/3) + C is there any ideas for a general case or the case n=2?

$${f}_{{n}} \left({x}\right):=\int{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \frac{{cos}\left({x}\right)}{\:\left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{{n}}{\mathrm{3}}} }{dx}=...? \\ $$$${for}\:{n}=\mathrm{1},\:{i}\:{found}\: \\ $$$$\:\:\:\:\:\:{f}_{\mathrm{1}} \left({x}\right)=\frac{\mathrm{3}}{\mathrm{4}}{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\:{C} \\ $$$${is}\:{there}\:{any}\:{ideas}\:{for}\:{a}\:{general}\:{case}\:{or} \\ $$$${the}\:{case}\:{n}=\mathrm{2}? \\ $$

Question Number 207423 Answers: 2 Comments: 0

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$$\:\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 207416 Answers: 2 Comments: 0

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Question Number 207395 Answers: 1 Comments: 0

Geometric series: ((b_4 ∙ b_7 ∙ b_(10) )/(b_1 ∙ b_3 ∙ b_5 )) = 2^(12) find: (b_5 /b_2 ) = ?

$$\mathrm{Geometric}\:\mathrm{series}: \\ $$$$\frac{\mathrm{b}_{\mathrm{4}} \:\centerdot\:\mathrm{b}_{\mathrm{7}} \:\centerdot\:\mathrm{b}_{\mathrm{10}} }{\mathrm{b}_{\mathrm{1}} \:\centerdot\:\mathrm{b}_{\mathrm{3}} \:\centerdot\:\mathrm{b}_{\mathrm{5}} }\:\:=\:\:\mathrm{2}^{\mathrm{12}} \:\:\:\:\:\mathrm{find}:\:\:\:\frac{\mathrm{b}_{\mathrm{5}} }{\mathrm{b}_{\mathrm{2}} }\:\:=\:\:? \\ $$

Question Number 207394 Answers: 1 Comments: 0

(a/b) = (c/d) a^3 − b^3 = 625 c^3 − d^3 = 1 Find: a,b,c,d = ?

$$\frac{\mathrm{a}}{\mathrm{b}}\:\:=\:\:\frac{\mathrm{c}}{\mathrm{d}} \\ $$$$\mathrm{a}^{\mathrm{3}} \:−\:\mathrm{b}^{\mathrm{3}} \:=\:\mathrm{625} \\ $$$$\mathrm{c}^{\mathrm{3}} \:−\:\mathrm{d}^{\mathrm{3}} \:=\:\mathrm{1} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:=\:? \\ $$

Question Number 207390 Answers: 0 Comments: 1

Question Number 207389 Answers: 0 Comments: 1

Question Number 207387 Answers: 1 Comments: 0

Let cardE=n , and the set of parts S={(A,B)∈P(E)×P(E) / A∩B=∅} Show that cardS= 3^n

$${Let}\:\:{cardE}={n}\:,\:{and}\:\:{the}\:{set}\:{of}\:{parts} \\ $$$${S}=\left\{\left({A},{B}\right)\in{P}\left({E}\right)×{P}\left({E}\right)\:/\:\:{A}\cap{B}=\varnothing\right\} \\ $$$${Show}\:{that}\:\:{cardS}=\:\mathrm{3}^{{n}} \\ $$

Question Number 207385 Answers: 0 Comments: 1

Find: (√((2,5−(√5))^2 )) − (((1,5−(√5))^3 )^(1/2) ))^(1/3) − (√2) sin ((7π)/4)

$$\mathrm{Find}: \\ $$$$\sqrt{\left(\mathrm{2},\mathrm{5}−\sqrt{\mathrm{5}}\right)^{\mathrm{2}} }\:−\:\sqrt[{\mathrm{3}}]{\left.\left(\mathrm{1},\mathrm{5}−\sqrt{\mathrm{5}}\right)^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} }\:−\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:\frac{\mathrm{7}\pi}{\mathrm{4}} \\ $$

Question Number 207383 Answers: 0 Comments: 1

∫((ln(x^2 +sin(sin(e^x ))))/( (√(x+tan(ln(x))))))dx

$$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$

Question Number 207382 Answers: 1 Comments: 0

Question Number 207381 Answers: 1 Comments: 0

Question Number 207374 Answers: 2 Comments: 0

Show that Σ_(k=0) ^n (C_n ^k )^2 =C_(2n) ^n

$${Show}\:{that}\:\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} ={C}_{\mathrm{2}{n}} ^{{n}} \\ $$

Question Number 207372 Answers: 0 Comments: 6

2 students are passing a test of n questions with the same chance to find each one Show the chance that they both don′t find a same question is ((3/4))^n

$$\mathrm{2}\:{students}\:{are}\:{passing}\: \\ $$$${a}\:{test}\:{of}\:\:{n}\:{questions}\:{with} \\ $$$${the}\:{same}\:{chance}\:{to}\:{find}\:{each}\:{one} \\ $$$${Show}\:\:{the}\:{chance}\:{that}\:{they}\:{both} \\ $$$$\:{don}'{t}\:{find}\:{a}\:{same}\:{question}\:{is}\:\:\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{{n}} \\ $$

Question Number 207402 Answers: 1 Comments: 0

f(x) + 2f((1/x)) = 3x. f ′(x) = ?

$${f}\left({x}\right)\:+\:\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{3}{x}. \\ $$$$\:{f}\:'\left({x}\right)\:=\:? \\ $$

Question Number 207362 Answers: 2 Comments: 0

Question Number 207361 Answers: 1 Comments: 0

y = ((tgx + ctgx)/8) , (0 ; (π/2)) Find: min(y) = ?

$$\mathrm{y}\:=\:\frac{\mathrm{tg}\boldsymbol{\mathrm{x}}\:\:+\:\:\mathrm{ctg}\boldsymbol{\mathrm{x}}}{\mathrm{8}}\:\:\:\:\:,\:\:\:\:\:\left(\mathrm{0}\:;\:\frac{\pi}{\mathrm{2}}\right) \\ $$$$\mathrm{Find}:\:\:\:\mathrm{min}\left(\mathrm{y}\right)\:=\:? \\ $$

Question Number 207352 Answers: 1 Comments: 3

calculate: ∫_(Π/4) ^(Π/2) ⌊cot(x)⌋ dx

$${calculate}: \\ $$$$\:\int_{\frac{\Pi}{\mathrm{4}}} ^{\frac{\Pi}{\mathrm{2}}} \lfloor{cot}\left({x}\right)\rfloor\:{dx} \\ $$

Question Number 207354 Answers: 0 Comments: 4

Question Number 207359 Answers: 1 Comments: 0

∫((ln(x^2 +sin(sin(e^x ))))/( (√(x+tan(ln(x))))))dx

$$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$

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