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Question Number 100590 Answers: 2 Comments: 0

∫_0 ^∞ (dx/((1+x^(18) )^2 ))

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{18}} \right)^{\mathrm{2}} } \\ $$

Question Number 100587 Answers: 2 Comments: 1

If the coefficients of x^k and x^(k+1) in the expansion (2+3x)^(19) are equal , what is the value of k ?

$$\mathrm{If}\:\mathrm{the}\:\mathrm{coefficients}\:\mathrm{of}\:{x}^{{k}} \:\mathrm{and}\:{x}^{{k}+\mathrm{1}} \:\mathrm{in}\:\mathrm{the}\: \\ $$$$\mathrm{expansion}\:\left(\mathrm{2}+\mathrm{3}{x}\right)^{\mathrm{19}} \:\mathrm{are}\:\mathrm{equal}\:,\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:? \\ $$

Question Number 100585 Answers: 0 Comments: 0

Given that G = {1,(x + yi),(x−yi)} form a group under complex multiplication, describe the locus of the point (x,y)

$$\:\mathrm{Given}\:\mathrm{that}\:\:{G}\:=\:\left\{\mathrm{1},\left({x}\:+\:{yi}\right),\left({x}−{yi}\right)\right\}\:\mathrm{form}\:\mathrm{a}\:\mathrm{group} \\ $$$$\mathrm{under}\:\mathrm{complex}\:\mathrm{multiplication},\:\mathrm{describe}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\left({x},{y}\right) \\ $$

Question Number 100584 Answers: 1 Comments: 0

∫i^i^(i......∞) dx

$$\int{i}^{{i}^{{i}......\infty} } {dx} \\ $$

Question Number 100583 Answers: 0 Comments: 0

A transformation f on a complex plane is defined by z′ = (1 +i)z −3 + 4i show that f is a simultitude with radius r and centre Ω to be determined. Determine to the invariant point under f.

$$\:\mathrm{A}\:\mathrm{transformation}\:{f}\:\mathrm{on}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{plane} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{z}'\:=\:\left(\mathrm{1}\:+{i}\right){z}\:−\mathrm{3}\:+\:\mathrm{4}{i} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{simultitude}\:\mathrm{with}\:\mathrm{radius}\:{r}\:\mathrm{and}\:\mathrm{centre} \\ $$$$\Omega\:\mathrm{to}\:\mathrm{be}\:\mathrm{determined}. \\ $$$$\mathrm{Determine}\:\mathrm{to}\:\mathrm{the}\:\mathrm{invariant}\:\mathrm{point}\:\mathrm{under}\:{f}. \\ $$

Question Number 100581 Answers: 0 Comments: 0

If α=((2π)/7) then prove that tanαtan2α+tan2αtan4α+tan4αtanα=−7

$${If}\:\alpha=\frac{\mathrm{2}\pi}{\mathrm{7}} \\ $$$${then}\:{prove}\:{that} \\ $$$${tan}\alpha{tan}\mathrm{2}\alpha+{tan}\mathrm{2}\alpha{tan}\mathrm{4}\alpha+{tan}\mathrm{4}\alpha{tan}\alpha=−\mathrm{7} \\ $$

Question Number 100575 Answers: 1 Comments: 0

Question Number 100570 Answers: 0 Comments: 1

Question Number 100567 Answers: 0 Comments: 3

{ ((x−(√(yz)) = 42)),((y−(√(xz)) = 6)),((z−(√(xy)) = −30)) :} find x+y+z =

$$\begin{cases}{{x}−\sqrt{{yz}}\:=\:\mathrm{42}}\\{{y}−\sqrt{{xz}}\:=\:\mathrm{6}}\\{{z}−\sqrt{{xy}}\:=\:−\mathrm{30}}\end{cases} \\ $$$${find}\:{x}+{y}+{z}\:= \\ $$

Question Number 100565 Answers: 1 Comments: 1

Question Number 100562 Answers: 0 Comments: 0

Question Number 100561 Answers: 1 Comments: 1

Question Number 100557 Answers: 2 Comments: 0

Ω=∫_0 ^∞ (e^(ax) /(e^(bx) +1))dx, b>a

$$\Omega=\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{e}^{{ax}} }{{e}^{{bx}} +\mathrm{1}}{dx},\:{b}>{a} \\ $$

Question Number 100538 Answers: 0 Comments: 1

Question Number 100540 Answers: 0 Comments: 1

Question Number 100539 Answers: 0 Comments: 2

(−1)^n Σ_(n=1) ^∞ (3^n /n)

$$\left(−\mathrm{1}\right)^{{n}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{3}^{{n}} }{{n}} \\ $$

Question Number 100543 Answers: 2 Comments: 1

Question Number 100522 Answers: 1 Comments: 0

Question Number 100514 Answers: 2 Comments: 0

calculatelim_(n→+∞) ∫_0 ^∞ (1−(x/n))^n ln(1+2x)dx

$$\mathrm{calculatelim}_{\mathrm{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\left(\mathrm{1}−\frac{\mathrm{x}}{\mathrm{n}}\right)^{\mathrm{n}} \mathrm{ln}\left(\mathrm{1}+\mathrm{2x}\right)\mathrm{dx} \\ $$

Question Number 100513 Answers: 0 Comments: 0

findA_(nm) =∫_0 ^∞ e^(−nx) ∣sin(px)∣ dx with n and p integr natural ≥1

$$\mathrm{findA}_{\mathrm{nm}} \:=\int_{\mathrm{0}} ^{\infty} \:\:\mathrm{e}^{−\mathrm{nx}} \:\mid\mathrm{sin}\left(\mathrm{px}\right)\mid\:\mathrm{dx}\:\:\mathrm{with}\:\:\mathrm{n}\:\mathrm{and}\:\mathrm{p}\:\mathrm{integr}\:\mathrm{natural}\:\geqslant\mathrm{1} \\ $$

Question Number 100512 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) (x^n /((x^2 +x+1)^n )) dx with n integr and n≥2

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{x}^{\mathrm{n}} }{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}\right)^{\mathrm{n}} }\:\mathrm{dx}\:\:\mathrm{with}\:\mathrm{n}\:\mathrm{integr}\:\mathrm{and}\:\mathrm{n}\geqslant\mathrm{2} \\ $$

Question Number 100511 Answers: 1 Comments: 0

calculate ∫_(−∞) ^∞ ((arctan(cosx +sinx))/(x^2 +4)) dx

$$\mathrm{calculate}\:\:\int_{−\infty} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{cosx}\:+\mathrm{sinx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}}\:\mathrm{dx} \\ $$

Question Number 100498 Answers: 0 Comments: 0

Question Number 100528 Answers: 1 Comments: 2

Question Number 100492 Answers: 2 Comments: 3

((16−((64)/(16−((64)/(16−((64)/(16−...))))))))^(1/(3 )) −((−2−(1/(−2−(1/(−2−(1/(−2−...))))))))^(1/(3 ))

$$\sqrt[{\mathrm{3}\:\:\:}]{\mathrm{16}−\frac{\mathrm{64}}{\mathrm{16}−\frac{\mathrm{64}}{\mathrm{16}−\frac{\mathrm{64}}{\mathrm{16}−...}}}}−\sqrt[{\mathrm{3}\:\:}]{−\mathrm{2}−\frac{\mathrm{1}}{−\mathrm{2}−\frac{\mathrm{1}}{−\mathrm{2}−\frac{\mathrm{1}}{−\mathrm{2}−...}}}} \\ $$

Question Number 100491 Answers: 1 Comments: 0

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