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Question Number 148819 Answers: 2 Comments: 0

Question Number 148818 Answers: 0 Comments: 0

∫_( 0) ^( ∞) ((xlog(x)log^3 (x+1))/((x+1)^3 )) = ?

$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{{xlog}\left({x}\right){log}^{\mathrm{3}} \left({x}+\mathrm{1}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} }\:=\:? \\ $$

Question Number 148756 Answers: 1 Comments: 0

{ ((x^2 + xy + x + y = −2)),((y^2 + xy + x + y = 1)) :} ⇒ 3x + y = ?

$$\begin{cases}{{x}^{\mathrm{2}} \:+\:{xy}\:+\:{x}\:+\:{y}\:=\:−\mathrm{2}}\\{{y}^{\mathrm{2}} \:+\:{xy}\:+\:{x}\:+\:{y}\:=\:\mathrm{1}}\end{cases}\:\:\Rightarrow\:\mathrm{3}{x}\:+\:{y}\:=\:? \\ $$

Question Number 148754 Answers: 1 Comments: 1

Question Number 148753 Answers: 1 Comments: 0

Question Number 148746 Answers: 1 Comments: 0

if x - y = y - z = 4 find x^2 + z^2 - 2y^2 = ?

$${if}\:\:\:{x}\:-\:{y}\:=\:{y}\:-\:{z}\:=\:\mathrm{4} \\ $$$${find}\:\:\:{x}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:-\:\mathrm{2}{y}^{\mathrm{2}} \:=\:? \\ $$

Question Number 148745 Answers: 0 Comments: 0

Question Number 150871 Answers: 1 Comments: 0

Find all the real solutions of cos x+cos^5 x+cos 7x=3

$$\:\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$\mathrm{of}\:\mathrm{cos}\:\mathrm{x}+\mathrm{cos}\:^{\mathrm{5}} \mathrm{x}+\mathrm{cos}\:\mathrm{7x}=\mathrm{3} \\ $$

Question Number 150870 Answers: 1 Comments: 1

Question Number 148724 Answers: 1 Comments: 0

find laurent series f(z)=(1/(z^2 −z+1)) ,0<∣z−1∣<1

$${find}\:{laurent}\:{series}\:{f}\left({z}\right)=\frac{\mathrm{1}}{{z}^{\mathrm{2}} −{z}+\mathrm{1}}\:\:,\mathrm{0}<\mid{z}−\mathrm{1}\mid<\mathrm{1} \\ $$

Question Number 148723 Answers: 0 Comments: 1

tgx+tgy=4 cosx+cosy=(1/5) tg(x+y)=?

$${tgx}+{tgy}=\mathrm{4}\:\:\:{cosx}+{cosy}=\frac{\mathrm{1}}{\mathrm{5}}\:\:\: \\ $$$${tg}\left({x}+{y}\right)=? \\ $$

Question Number 148720 Answers: 4 Comments: 0

Find the Talor series of ((ln(1−x))/((1−x)^2 )) at x=0.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{Talor}\:\mathrm{series}\:\mathrm{of}\:\frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{at}\:\mathrm{x}=\mathrm{0}. \\ $$

Question Number 148711 Answers: 2 Comments: 0

if x+(1/x)=4, and x−(1/x)=3 then prove that 4=5

$${if}\:{x}+\frac{\mathrm{1}}{{x}}=\mathrm{4},\:{and}\:{x}−\frac{\mathrm{1}}{{x}}=\mathrm{3} \\ $$$${then}\:{prove}\:{that}\:\mathrm{4}=\mathrm{5} \\ $$

Question Number 148707 Answers: 1 Comments: 0

Solve for equation: 2tg(3x) - 3tg(2x) = tg^2 (2x) ∙ tg(3x)

$${Solve}\:{for}\:{equation}: \\ $$$$\mathrm{2}{tg}\left(\mathrm{3}{x}\right)\:-\:\mathrm{3}{tg}\left(\mathrm{2}{x}\right)\:=\:{tg}^{\mathrm{2}} \left(\mathrm{2}{x}\right)\:\centerdot\:{tg}\left(\mathrm{3}{x}\right) \\ $$

Question Number 148706 Answers: 1 Comments: 0

Question Number 148708 Answers: 0 Comments: 1

tg^6 (36°) + tg^6 (72°) = ?

$${tg}^{\mathrm{6}} \left(\mathrm{36}°\right)\:+\:{tg}^{\mathrm{6}} \left(\mathrm{72}°\right)\:=\:? \\ $$

Question Number 148694 Answers: 0 Comments: 3

lim_(x→2) ((x^2 −2^x )/( ((x^x +x))^(1/5) −(6)^(1/5) )) =?

$$\:\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{2}^{\mathrm{x}} }{\:\sqrt[{\mathrm{5}}]{\mathrm{x}^{\mathrm{x}} +\mathrm{x}}−\sqrt[{\mathrm{5}}]{\mathrm{6}}}\:=? \\ $$

Question Number 148739 Answers: 1 Comments: 2

Question Number 148689 Answers: 2 Comments: 0

Solve for x∈R ((x^2 −x+1)/(x^2 +x+1)) = (√((x^3 −1)/(x^3 +1))) .

$$\:\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\in\mathrm{R} \\ $$$$\:\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\:=\:\sqrt{\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{1}}{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}}\:. \\ $$

Question Number 148733 Answers: 0 Comments: 0

tg(α)+tg(β)=4 cos(α)+cos(β)=(1/5) tg(α+β)=?

$${tg}\left(\alpha\right)+{tg}\left(\beta\right)=\mathrm{4} \\ $$$${cos}\left(\alpha\right)+{cos}\left(\beta\right)=\frac{\mathrm{1}}{\mathrm{5}} \\ $$$${tg}\left(\alpha+\beta\right)=? \\ $$

Question Number 148732 Answers: 3 Comments: 0

show that 8^n −3^n is divisible by 5 for all natural number.

$$\:\:\:\:\:\:{show}\:{that}\: \\ $$$$\mathrm{8}^{{n}} \:−\mathrm{3}^{{n}} \:{is}\:{divisible}\:{by}\:\mathrm{5}\:\:{for}\:{all}\:{natural} \\ $$$${number}. \\ $$

Question Number 148677 Answers: 1 Comments: 0

A series of natural numbers are grouped as 1+(2+3)+(4+5+6)+... such that the rth group contains r terms. Show that the sum of the numbers in the (2r−1)th group is r^4 −(r−1)^4 .

$$\:\:\mathrm{A}\:\mathrm{series}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{are}\: \\ $$$$\:\:\mathrm{grouped}\:\mathrm{as}\:\mathrm{1}+\left(\mathrm{2}+\mathrm{3}\right)+\left(\mathrm{4}+\mathrm{5}+\mathrm{6}\right)+... \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:{rth}\:\mathrm{group}\:\mathrm{contains}\:{r}\: \\ $$$$\:\:\mathrm{terms}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\:\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{the}\:\left(\mathrm{2}{r}−\mathrm{1}\right){th}\:\mathrm{group}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{r}}^{\mathrm{4}} −\left(\boldsymbol{{r}}−\mathrm{1}\right)^{\mathrm{4}} . \\ $$

Question Number 148676 Answers: 1 Comments: 0

Find local minimum of function H(x)=((34)/(3+((2x)/(x^2 +3x+1)))) .

$$\mathrm{Find}\:\mathrm{local}\:\mathrm{minimum}\:\mathrm{of}\:\mathrm{function} \\ $$$$\:\mathrm{H}\left(\mathrm{x}\right)=\frac{\mathrm{34}}{\mathrm{3}+\frac{\mathrm{2x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{3x}+\mathrm{1}}}\:. \\ $$

Question Number 148674 Answers: 1 Comments: 0

Find max & min value of (1) f(x)=sin^3 x (√(1+cos^2 x)) +cos^3 x(√(1+sin^2 x)) . (2) f(x)=sin x (√(1+cos^2 x)) +cos x (√(1+sin^2 x)) . x ∈ R

$$\:\mathrm{Find}\:\mathrm{max}\:\&\:\mathrm{min}\:\mathrm{value}\:\mathrm{of} \\ $$$$\:\left(\mathrm{1}\right)\:\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}\:+\mathrm{cos}\:^{\mathrm{3}} \mathrm{x}\sqrt{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}\:. \\ $$$$\left(\mathrm{2}\right)\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{sin}\:\mathrm{x}\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}\:+\mathrm{cos}\:\mathrm{x}\:\sqrt{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}\:. \\ $$$$\:\mathrm{x}\:\in\:\mathbb{R}\: \\ $$

Question Number 148655 Answers: 3 Comments: 0

lim_(n→∞) ∫_( 0) ^( 1) ((nx)/(1 + n^2 x^4 )) dx = ?

$$\underset{\boldsymbol{{n}}\rightarrow\infty} {{lim}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{{nx}}{\mathrm{1}\:+\:{n}^{\mathrm{2}} {x}^{\mathrm{4}} }\:{dx}\:=\:? \\ $$

Question Number 148645 Answers: 1 Comments: 0

u_(n+3) =((u_(n+2) +u_(n+1) +u_n )/3) , ∀n∈IN find u_n

$${u}_{{n}+\mathrm{3}} =\frac{{u}_{{n}+\mathrm{2}} +{u}_{{n}+\mathrm{1}} +{u}_{{n}} }{\mathrm{3}}\:,\:\forall{n}\in{IN} \\ $$$${find}\:{u}_{{n}} \: \\ $$

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