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Question Number 151561 Answers: 1 Comments: 0

Given that x+iy=(a/(b+sin θ+icos θ)) show that (b^2 −1)(x^2 +y^2 )+a^2 =2abx

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{x}+\mathrm{iy}=\frac{\mathrm{a}}{\mathrm{b}+\mathrm{sin}\:\theta+\mathrm{icos}\:\theta} \\ $$$$\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{b}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)+\mathrm{a}^{\mathrm{2}} =\mathrm{2abx} \\ $$

Question Number 151560 Answers: 4 Comments: 0

show that i^i is always real

$$\mathrm{show}\:\mathrm{that}\:\:\mathrm{i}^{\mathrm{i}} \:\:\mathrm{is}\:\mathrm{always}\:\mathrm{real} \\ $$

Question Number 151559 Answers: 2 Comments: 0

prove that (1+cos θ+isin θ)^n + (1+cos θ−isin θ)^n =2^(n+1) cos (θ/2)cos ((nθ)/2)

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}+\mathrm{cos}\:\theta+\mathrm{isin}\:\theta\right)^{\mathrm{n}} \\ $$$$+\:\left(\mathrm{1}+\mathrm{cos}\:\theta−\mathrm{isin}\:\theta\right)^{\mathrm{n}} =\mathrm{2}^{\mathrm{n}+\mathrm{1}} \mathrm{cos}\:\frac{\theta}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{n}\theta}{\mathrm{2}} \\ $$

Question Number 151554 Answers: 1 Comments: 0

Question Number 151549 Answers: 1 Comments: 0

Compare: 2^2^2^.^.^. and 3^3^3^.^.^. Here it is raised 1001 times a square, 1000 times a cube.

$$\mathrm{Compare}: \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{.^{.^{.} } } } } \:\:\:\:\:\mathrm{and}\:\:\:\:\:\mathrm{3}^{\mathrm{3}^{\mathrm{3}^{.^{.^{.} } } } } \\ $$$$\mathrm{Here}\:\mathrm{it}\:\mathrm{is}\:\mathrm{raised}\:\mathrm{1001}\:\mathrm{times}\:\mathrm{a}\:\mathrm{square}, \\ $$$$\mathrm{1000}\:\mathrm{times}\:\mathrm{a}\:\mathrm{cube}. \\ $$

Question Number 151538 Answers: 3 Comments: 0

Question Number 151533 Answers: 0 Comments: 10

Compare: 2^2^2 and 3^3^3

$$\mathrm{Compare}: \\ $$$$\mathrm{2}^{\mathrm{2}^{\mathrm{2}} } \:\:\:\mathrm{and}\:\:\:\mathrm{3}^{\mathrm{3}^{\mathrm{3}} } \\ $$

Question Number 151531 Answers: 0 Comments: 1

Question Number 151519 Answers: 0 Comments: 0

∫_0 ^( ∞) ((ln(⌊x^2 ⌋!))/( (x^x +1)^x )) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\left(\lfloor{x}^{\mathrm{2}} \rfloor!\right)}{\:\left({x}^{{x}} +\mathrm{1}\right)^{{x}} }\:{dx} \\ $$$$\: \\ $$

Question Number 151518 Answers: 0 Comments: 0

Question Number 151513 Answers: 2 Comments: 0

Find two possible values of p if the lines px−y=0 and 3x+y+1=0 intersect at 45°

$$\mathrm{Find}\:\mathrm{two}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:{p}\:\mathrm{if}\:\mathrm{the}\:\mathrm{lines} \\ $$$${px}−{y}=\mathrm{0}\:\mathrm{and}\:\mathrm{3}{x}+{y}+\mathrm{1}=\mathrm{0}\:\mathrm{intersect}\:\mathrm{at}\:\mathrm{45}° \\ $$

Question Number 151504 Answers: 1 Comments: 0

∫_0 ^( ∞) ((ln x)/( (√x) (√(x+1)) (√(2x+1)))) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\:{x}}{\:\sqrt{{x}}\:\sqrt{{x}+\mathrm{1}}\:\sqrt{\mathrm{2}{x}+\mathrm{1}}}\:{dx} \\ $$$$\: \\ $$

Question Number 151503 Answers: 1 Comments: 0

∫_0 ^( ∞) ((x−1)/( (√(2^x −1)) ln(2^x −1))) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{x}−\mathrm{1}}{\:\sqrt{\mathrm{2}^{{x}} −\mathrm{1}}\:\mathrm{ln}\left(\mathrm{2}^{{x}} −\mathrm{1}\right)}\:{dx} \\ $$$$\: \\ $$

Question Number 151501 Answers: 1 Comments: 0

∫ (log x + 1)x^x dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\left(\mathrm{log}\:{x}\:+\:\mathrm{1}\right){x}^{{x}} \:{dx} \\ $$$$\: \\ $$

Question Number 151496 Answers: 1 Comments: 0

∀x,y∈R_+ ^∗ , show that (x/(x^4 +y^2 ))+(y/(y^4 +x^2 ))≤(1/(xy))

$$\forall{x},{y}\in\mathbb{R}_{+} ^{\ast} ,\:{show}\:{that}\:\frac{{x}}{{x}^{\mathrm{4}} +{y}^{\mathrm{2}} }+\frac{{y}}{{y}^{\mathrm{4}} +{x}^{\mathrm{2}} }\leqslant\frac{\mathrm{1}}{{xy}} \\ $$

Question Number 151495 Answers: 0 Comments: 4

Question Number 151493 Answers: 0 Comments: 2

((cos (((5π)/2)−6x)+sin (π+4x)+sin (3π−x))/(sin (((5π)/2)+6x)+cos (4x−2π)+cos (x+2π))) =?

$$\:\:\:\frac{\mathrm{cos}\:\left(\frac{\mathrm{5}\pi}{\mathrm{2}}−\mathrm{6}{x}\right)+\mathrm{sin}\:\left(\pi+\mathrm{4}{x}\right)+\mathrm{sin}\:\left(\mathrm{3}\pi−{x}\right)}{\mathrm{sin}\:\left(\frac{\mathrm{5}\pi}{\mathrm{2}}+\mathrm{6}{x}\right)+\mathrm{cos}\:\left(\mathrm{4}{x}−\mathrm{2}\pi\right)+\mathrm{cos}\:\left({x}+\mathrm{2}\pi\right)}\:=? \\ $$

Question Number 151492 Answers: 0 Comments: 0

∫_0 ^( 1) ((tan(x^2 +1))/(x^2 +1)) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{tan}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$$$\: \\ $$

Question Number 151490 Answers: 1 Comments: 0

{ ((a^2 =2a+b)),((b^2 =a+2b)) :} and a≠b find (√(a^2 +b^2 +1))

$$\begin{cases}{\mathrm{a}^{\mathrm{2}} =\mathrm{2a}+\mathrm{b}}\\{\mathrm{b}^{\mathrm{2}} =\mathrm{a}+\mathrm{2b}}\end{cases}\:\:\mathrm{and}\:\:\mathrm{a}\neq\mathrm{b}\:\:\mathrm{find}\:\:\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 151486 Answers: 1 Comments: 0

∫ln(sinx)=? x ≠ 0 + 2kπ ; k ∈ Z

$$\int{ln}\left({sinx}\right)=?\: \\ $$$${x}\:\neq\:\mathrm{0}\:+\:\mathrm{2}{k}\pi\:;\:{k}\:\in\:\mathbb{Z} \\ $$

Question Number 151485 Answers: 1 Comments: 0

Question Number 151481 Answers: 0 Comments: 2

if x;y;z;t∈R and x+y+z ≤ 3t ; y+z+t ≤ 3x z+t+x ≤ 3y ; t+x+y ≤ 3z compare: x;y;z;t

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}\in\mathbb{R}\:\:\mathrm{and} \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:\leqslant\:\mathrm{3t}\:\:;\:\:\mathrm{y}+\mathrm{z}+\mathrm{t}\:\leqslant\:\mathrm{3x} \\ $$$$\mathrm{z}+\mathrm{t}+\mathrm{x}\:\leqslant\:\mathrm{3y}\:\:;\:\:\mathrm{t}+\mathrm{x}+\mathrm{y}\:\leqslant\:\mathrm{3z} \\ $$$$\mathrm{compare}:\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t} \\ $$

Question Number 151636 Answers: 1 Comments: 4

Question Number 151475 Answers: 2 Comments: 0

if x∈R prove that: x^6 - x + 1 > 0

$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}\in\mathbb{R}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{x}^{\mathrm{6}} \:-\:\mathrm{x}\:+\:\mathrm{1}\:>\:\mathrm{0} \\ $$

Question Number 151468 Answers: 2 Comments: 0

z^2 −7z+16=i(z−11)

$$\mathrm{z}^{\mathrm{2}} −\mathrm{7z}+\mathrm{16}=\mathrm{i}\left(\mathrm{z}−\mathrm{11}\right) \\ $$$$ \\ $$

Question Number 151461 Answers: 1 Comments: 0

∫((xdx)/(x^4 +4x^2 +5))

$$\int\frac{\mathrm{xdx}}{\mathrm{x}^{\mathrm{4}} +\mathrm{4x}^{\mathrm{2}} +\mathrm{5}} \\ $$

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