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Question Number 155686 Answers: 2 Comments: 0

∫_0 ^(π/2) (dx/(1+tan x)) =?

$$\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{dx}}{\mathrm{1}+\mathrm{tan}\:{x}}\:=? \\ $$

Question Number 155677 Answers: 1 Comments: 1

Question Number 155674 Answers: 0 Comments: 0

monster integral ∫_0 ^( (π/4)) ln^2 (sin(2x)+ cos(3x)) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underline{\mathrm{monster}\:\mathrm{integral}} \\ $$$$\: \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{sin}\left(\mathrm{2}{x}\right)+\:\mathrm{cos}\left(\mathrm{3}{x}\right)\right)\:{dx} \\ $$$$\: \\ $$$$\: \\ $$

Question Number 155673 Answers: 0 Comments: 0

Question Number 155667 Answers: 0 Comments: 2

Question Number 155657 Answers: 2 Comments: 1

Question Number 155653 Answers: 0 Comments: 0

if a;b;c≥0 and a+b+c=1 prove that 18 Σ ab + 45 Σ a^2 b ≤ 11

$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\geqslant\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1}\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{18}\:\Sigma\:\mathrm{ab}\:+\:\mathrm{45}\:\Sigma\:\mathrm{a}^{\mathrm{2}} \mathrm{b}\:\leqslant\:\mathrm{11} \\ $$

Question Number 155650 Answers: 2 Comments: 0

Solve for real numbers: (√((x-a)/(x-b))) + (a/x) = (√((x-b)/(x-a))) + (b/x) a;b∈R and a≠b

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt{\frac{\mathrm{x}-\mathrm{a}}{\mathrm{x}-\mathrm{b}}}\:+\:\frac{\mathrm{a}}{\mathrm{x}}\:=\:\sqrt{\frac{\mathrm{x}-\mathrm{b}}{\mathrm{x}-\mathrm{a}}}\:+\:\frac{\mathrm{b}}{\mathrm{x}} \\ $$$$\mathrm{a};\mathrm{b}\in\mathbb{R}\:\:\mathrm{and}\:\:\mathrm{a}\neq\mathrm{b} \\ $$

Question Number 155643 Answers: 0 Comments: 1

Question Number 155642 Answers: 0 Comments: 3

Question Number 155639 Answers: 1 Comments: 4

Question Number 155638 Answers: 1 Comments: 2

Question Number 155631 Answers: 2 Comments: 2

(x+3(x−2)=x+10

$$\left({x}+\mathrm{3}\left({x}−\mathrm{2}\right)={x}+\mathrm{10}\right. \\ $$

Question Number 155630 Answers: 1 Comments: 0

Question Number 155628 Answers: 1 Comments: 0

Question Number 155625 Answers: 2 Comments: 0

lim_(x→0) (((tanx)/x))^(1/x^2 ) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{{tanx}}{{x}}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} =\:? \\ $$

Question Number 155621 Answers: 1 Comments: 0

Question Number 155620 Answers: 1 Comments: 0

Find: 𝛀 = lim_(x→1) ((∫_(x-1) ^(e^x -e) cos(t^5 )dt)/(3^x - 3)) = ?

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\underset{\boldsymbol{\mathrm{x}}-\mathrm{1}} {\overset{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} -\boldsymbol{\mathrm{e}}} {\int}}\mathrm{cos}\left(\mathrm{t}^{\mathrm{5}} \right)\mathrm{dt}}{\mathrm{3}^{\boldsymbol{\mathrm{x}}} \:-\:\mathrm{3}}\:=\:? \\ $$

Question Number 155619 Answers: 1 Comments: 0

𝛀 =∫_( 0) ^( 1) (x^(49) /(1 + x + x^2 + x^3 ... x^(100) )) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{49}} }{\mathrm{1}\:+\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{3}} \:...\:\mathrm{x}^{\mathrm{100}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 155618 Answers: 0 Comments: 0

if x;y∈R and x^3 +y^3 =16 prove that: x^4 + y^4 + 2x^2 + y^2 ≥ 4x + 36

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}\in\mathbb{R}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} =\mathrm{16}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:\geqslant\:\mathrm{4x}\:+\:\mathrm{36} \\ $$

Question Number 155604 Answers: 1 Comments: 0

Question Number 155594 Answers: 1 Comments: 0

Question Number 155586 Answers: 1 Comments: 0

How to proof f:X→Y f is 1 to 1 ⇐⇒ f(E)\f(F)=f(E\F)

$${How}\:{to}\:{proof}\:\:\:\:\:{f}:{X}\rightarrow{Y} \\ $$$${f}\:{is}\:\mathrm{1}\:{to}\:\mathrm{1}\:\Leftarrow\Rightarrow\:{f}\left({E}\right)\backslash{f}\left({F}\right)={f}\left({E}\backslash{F}\right) \\ $$

Question Number 155585 Answers: 0 Comments: 0

Evaluate the limit and prove by the ε−δ definition that as n→∞ for z≥1 (2(z)^(1/n) − 1)^n → z^2

$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{by}\:\mathrm{the} \\ $$$$\varepsilon−\delta\:\mathrm{definition}\:\mathrm{that}\:\mathrm{as}\:\mathrm{n}\rightarrow\infty\:\mathrm{for}\:\mathrm{z}\geqslant\mathrm{1} \\ $$$$\left(\mathrm{2}\sqrt[{\boldsymbol{\mathrm{n}}}]{\mathrm{z}}\:−\:\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \:\rightarrow\:\mathrm{z}^{\mathrm{2}} \\ $$

Question Number 155583 Answers: 2 Comments: 0

Question Number 155576 Answers: 7 Comments: 3

the polynomial 4x^3 +ax^2 +bx+9, where a and b consant, is denoted by f(x). when f(x) is divide by (x−2) the remainder is r and when divided by (x−3) the remaider is 6r. its further given that (x+3) is a factor of f(x). Show that b−a=14 and hence find a and b

$${the}\:{polynomial}\:\mathrm{4}{x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+\mathrm{9}, \\ $$$${where}\:\boldsymbol{{a}}\:\:{and}\:\:\boldsymbol{{b}}\:{consant},\:{is}\:{denoted}\:{by} \\ $$$$\:{f}\left({x}\right).\:{when}\:{f}\left({x}\right)\:{is}\:{divide}\:{by}\:\left({x}−\mathrm{2}\right)\:{the} \\ $$$${remainder}\:{is}\:\boldsymbol{{r}}\:{and}\:{when}\:{divided}\:{by} \\ $$$$\:\left({x}−\mathrm{3}\right)\:{the}\:{remaider}\:{is}\:\mathrm{6}\boldsymbol{{r}}.\:{its}\:{further} \\ $$$$\:{given}\:{that}\:\left({x}+\mathrm{3}\right)\:{is}\:{a}\:{factor}\:{of}\:{f}\left({x}\right). \\ $$$$\:{Show}\:{that}\:\boldsymbol{{b}}−\boldsymbol{{a}}=\mathrm{14} \\ $$$$\:{and}\:{hence}\:{find}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}} \\ $$

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