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Question Number 83977 Answers: 2 Comments: 0

find range function f(x)= x(√(7x−x^2 −1)) without calculus

$$\mathrm{find}\:\mathrm{range}\:\mathrm{function}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{x}\sqrt{\mathrm{7x}−\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\:\mathrm{without} \\ $$$$\mathrm{calculus} \\ $$

Question Number 83975 Answers: 2 Comments: 0

find solution 8 tan x−8tan^5 x = sec^6 x in x∈ (0, (π/2))

$$\mathrm{find}\:\mathrm{solution} \\ $$$$\mathrm{8}\:\mathrm{tan}\:\mathrm{x}−\mathrm{8tan}\:^{\mathrm{5}} \mathrm{x}\:=\:\mathrm{sec}\:^{\mathrm{6}} \mathrm{x}\:\mathrm{in}\:\mathrm{x}\in\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right) \\ $$

Question Number 83966 Answers: 2 Comments: 0

prove that for any complex number z, if ∣z∣ < 1, then Re(z + 1) > 0

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{complex}\:\mathrm{number}\:{z},\:\mathrm{if}\: \\ $$$$\:\mid{z}\mid\:<\:\mathrm{1},\:\mathrm{then}\:\mathrm{Re}\left({z}\:+\:\mathrm{1}\right)\:>\:\mathrm{0} \\ $$

Question Number 83965 Answers: 0 Comments: 3

prove or disprove(with counter−example) that a) For all two dimensional vectors a,b,c, a.b = a. c ⇒ b=c. b) For all positive real numbers a,b. ((a +b)/2) ≥ (√(ab))

Question Number 83964 Answers: 1 Comments: 0

The graph of y = ((a + bx)/((x−1)(x−4))) has a turning point at P(2,−1). Find the value of a and b and hence,sketch the curve y = f(x) showing clearly the turning points, asympototes and intercept(s) with the axes.

$$\mathrm{The}\:\mathrm{graph}\:\mathrm{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:=\:\frac{{a}\:+\:{bx}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{4}\right)} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{turning}\:\mathrm{point}\:\mathrm{at}\:{P}\left(\mathrm{2},−\mathrm{1}\right).\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}\:\mathrm{and}\:{b}\: \\ $$$$\mathrm{and}\:\mathrm{hence},\mathrm{sketch}\:\mathrm{the}\:\mathrm{curve}\:{y}\:=\:{f}\left({x}\right)\:\mathrm{showing}\:\mathrm{clearly}\:\mathrm{the} \\ $$$$\mathrm{turning}\:\mathrm{points},\:\mathrm{asympototes}\:\mathrm{and}\:\mathrm{intercept}\left(\mathrm{s}\right)\:\mathrm{with}\:\mathrm{the} \\ $$$$\mathrm{axes}. \\ $$

Question Number 83961 Answers: 0 Comments: 1

find I =∫ e^(−x) cos^4 xdx and J =∫ e^(−x) sin^4 xdx

$${find}\:{I}\:=\int\:{e}^{−{x}} \:{cos}^{\mathrm{4}} {xdx}\:\:{and}\:{J}\:=\int\:{e}^{−{x}} \:{sin}^{\mathrm{4}} \:{xdx} \\ $$

Question Number 83960 Answers: 2 Comments: 1

∫ ((x−1)/(√(x^2 −x))) dx ?

$$\int\:\frac{\mathrm{x}−\mathrm{1}}{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}}}\:\mathrm{dx}\:?\: \\ $$

Question Number 83956 Answers: 1 Comments: 1

for x ∈ R satisfy the equation f(x)+3x f((1/x)) = 2(x+1) find f(2019) .

$$\mathrm{for}\:\mathrm{x}\:\in\:\mathbb{R}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{3x}\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:\mathrm{2}\left(\mathrm{x}+\mathrm{1}\right) \\ $$$$\mathrm{find}\:\mathrm{f}\left(\mathrm{2019}\right)\:.\: \\ $$

Question Number 83943 Answers: 1 Comments: 0

lim_(x→0) ((sec 6x−cos 2x)/(2x tan 5x))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:\mathrm{6x}−\mathrm{cos}\:\mathrm{2x}}{\mathrm{2x}\:\mathrm{tan}\:\mathrm{5x}} \\ $$$$ \\ $$

Question Number 83941 Answers: 1 Comments: 0

If ((2 ))^(1/3) + (4)^(1/(3 )) + ((8 ))^(1/(3 )) = x then x^3 −6x^2 +6x+6 = ?

$$\mathrm{If}\:\sqrt[{\mathrm{3}}]{\mathrm{2}\:}\:+\:\sqrt[{\mathrm{3}\:}]{\mathrm{4}}\:+\:\sqrt[{\mathrm{3}\:}]{\mathrm{8}\:}\:=\:\mathrm{x}\: \\ $$$$\mathrm{then}\:\mathrm{x}^{\mathrm{3}} −\mathrm{6x}^{\mathrm{2}} +\mathrm{6x}+\mathrm{6}\:=\:? \\ $$

Question Number 83927 Answers: 1 Comments: 0

∫((sinh(x)+e^(3x) )/(sinh(x)−e^x )) dx

$$\int\frac{{sinh}\left({x}\right)+{e}^{\mathrm{3}{x}} }{{sinh}\left({x}\right)−{e}^{{x}} }\:{dx} \\ $$

Question Number 83923 Answers: 1 Comments: 0

∫ ((x^2 +20)/((xsin x+5cos x)^2 )) dx = ?

$$\int\:\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{20}}{\left(\mathrm{xsin}\:\mathrm{x}+\mathrm{5cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 83921 Answers: 1 Comments: 0

∫ ((2x^(12) +5x^9 )/((x^5 +x^3 +1)^3 )) dx = ?

$$\int\:\:\frac{\mathrm{2x}^{\mathrm{12}} +\mathrm{5x}^{\mathrm{9}} }{\left(\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 83919 Answers: 0 Comments: 1

lim_(x→+∞) ((3/π)arc tan x)^(2x) = ?

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{3}}{\pi}\mathrm{arc}\:\mathrm{tan}\:{x}\right)^{\mathrm{2}{x}} \:=\:? \\ $$

Question Number 83917 Answers: 5 Comments: 0

find ∫ (dx/((1+(√(1+x^2 )))^2 ))

$${find}\:\int\:\:\:\frac{{dx}}{\left(\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{\mathrm{2}} } \\ $$

Question Number 83931 Answers: 1 Comments: 1

(1/(((√1)+(√2))((1)^(1/(4 )) +(2)^(1/(4 )) ))) + (1/(((√2)+(√3))((2)^(1/( 4)) +(3)^(1/(4 )) ))) + (1/(((√3)+(√4))((3)^(1/(4 )) +(4)^(1/(4 )) ))) + ... + (1/(((√(255))+(√(256)))(((255))^(1/(4 )) +((256))^(1/(4 )) ))) = ...

$$\frac{\mathrm{1}}{\left(\sqrt{\mathrm{1}}+\sqrt{\mathrm{2}}\right)\left(\sqrt[{\mathrm{4}\:}]{\mathrm{1}}+\sqrt[{\mathrm{4}\:}]{\mathrm{2}}\right)}\:+\:\frac{\mathrm{1}}{\left(\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}\right)\left(\sqrt[{\:\mathrm{4}}]{\mathrm{2}}+\sqrt[{\mathrm{4}\:}]{\mathrm{3}}\right)}\:+ \\ $$$$\frac{\mathrm{1}}{\left(\sqrt{\mathrm{3}}+\sqrt{\mathrm{4}}\right)\left(\sqrt[{\mathrm{4}\:}]{\mathrm{3}}+\sqrt[{\mathrm{4}\:}]{\mathrm{4}}\right)}\:+\:...\:+\:\frac{\mathrm{1}}{\left(\sqrt{\mathrm{255}}+\sqrt{\mathrm{256}}\right)\left(\sqrt[{\mathrm{4}\:}]{\mathrm{255}}+\sqrt[{\mathrm{4}\:}]{\mathrm{256}}\right)} \\ $$$$=\:...\: \\ $$

Question Number 83910 Answers: 2 Comments: 1

find all 6 digit numbers which are not only palindrome but also divisible by 495.

$$\mathrm{find}\:\mathrm{all}\:\mathrm{6}\:\mathrm{digit}\:\mathrm{numbers}\:\mathrm{which}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{only}\:\mathrm{palindrome}\:\mathrm{but}\:\mathrm{also}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{495}. \\ $$

Question Number 83899 Answers: 0 Comments: 1

Defined a function f(x) such that f(1−x)+2f(x)= nx for m ,n > 1 , the value of ∫ _1 ^( m) (2n+6f((m/x))) dx is ...

$$\mathrm{Defined}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{such}\: \\ $$$$\mathrm{that}\:\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)+\mathrm{2f}\left(\mathrm{x}\right)=\:\mathrm{nx}\: \\ $$$$\mathrm{for}\:\mathrm{m}\:,\mathrm{n}\:>\:\mathrm{1}\:,\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\int\underset{\mathrm{1}} {\overset{\:\mathrm{m}} {\:}}\left(\mathrm{2n}+\mathrm{6f}\left(\frac{\mathrm{m}}{\mathrm{x}}\right)\right)\:\mathrm{dx}\:\mathrm{is}\:... \\ $$

Question Number 83898 Answers: 0 Comments: 4

∫(du/((√(u^2 −1 ))−u))

$$\int\frac{\mathrm{du}}{\sqrt{\mathrm{u}^{\mathrm{2}} −\mathrm{1}\:}−\mathrm{u}} \\ $$

Question Number 83893 Answers: 0 Comments: 1

∫(du/(u−u^2 ))

$$\int\frac{\mathrm{du}}{\mathrm{u}−\mathrm{u}^{\mathrm{2}} } \\ $$

Question Number 83886 Answers: 0 Comments: 15

To the developers of TinkuTara: problem 1: i get no notifications when my posts are updated. problem 2: i can edit my post, see picture 1, but the content is not visiable, see picture 2.

$${To}\:{the}\:{developers}\:{of}\:{TinkuTara}: \\ $$$${problem}\:\mathrm{1}: \\ $$$${i}\:{get}\:{no}\:{notifications}\:{when}\:{my}\:{posts} \\ $$$${are}\:{updated}. \\ $$$$ \\ $$$${problem}\:\mathrm{2}: \\ $$$${i}\:{can}\:{edit}\:{my}\:{post},\:{see}\:{picture}\:\mathrm{1},\:{but} \\ $$$${the}\:{content}\:{is}\:{not}\:{visiable},\:{see}\:{picture}\:\mathrm{2}. \\ $$

Question Number 83935 Answers: 0 Comments: 0

If x^4 and higher powers of x are neglected, show that (√((((1−x)/(1+x+x^2 )))=1−x+(1/2)x^3 ))

$${If}\:\:\boldsymbol{{x}}^{\mathrm{4}} \:{and}\:{higher}\:{powers}\:{of}\:{x}\:{are}\:{neglected},\:{show}\:{that} \\ $$$$\sqrt{\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\right)=\mathrm{1}−{x}+\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{3}} } \\ $$

Question Number 83876 Answers: 0 Comments: 0

Question Number 83874 Answers: 1 Comments: 1

Question Number 83871 Answers: 0 Comments: 4

If equation { (((√(x^2 +y^2 ))+(√((x−4)^2 +y^2 ))+(√(x^2 +(y−3)^2 ))+(√((x−4)^2 +(y−3)^2 ))=10)),((x+2y= 5z)) :} has solution is (a,b,c). find a+2b+3c

$$\mathrm{If}\:\mathrm{equation}\: \\ $$$$\begin{cases}{\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }+\sqrt{\left(\mathrm{x}−\mathrm{4}\right)^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }+\sqrt{\mathrm{x}^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{3}\right)^{\mathrm{2}} }+\sqrt{\left(\mathrm{x}−\mathrm{4}\right)^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{3}\right)^{\mathrm{2}} }=\mathrm{10}}\\{\mathrm{x}+\mathrm{2y}=\:\mathrm{5z}}\end{cases} \\ $$$$\mathrm{has}\:\mathrm{solution}\:\mathrm{is}\:\left(\mathrm{a},\mathrm{b},\mathrm{c}\right).\: \\ $$$$\mathrm{find}\:\mathrm{a}+\mathrm{2b}+\mathrm{3c}\: \\ $$

Question Number 83865 Answers: 0 Comments: 3

lim_(x→−∞ ) (x(√(2x+2))−x(√(2x+3)))

$$\underset{{x}\rightarrow−\infty\:} {\mathrm{lim}}\:\left(\mathrm{x}\sqrt{\mathrm{2x}+\mathrm{2}}−\mathrm{x}\sqrt{\mathrm{2x}+\mathrm{3}}\right) \\ $$

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