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Question Number 161133    Answers: 1   Comments: 0

lim_(nβ†’βˆž) (1/n)∫_0 ^1 (dx/(x(x+(1/n))))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{n}}\right)}=? \\ $$

Question Number 161123    Answers: 0   Comments: 0

Find: 𝛀 =lim_(nβ†’βˆž) (H_n /(n(H_(2n-1) - 2 H_(n-1) )))

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{H}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}\left(\mathrm{H}_{\mathrm{2}\boldsymbol{\mathrm{n}}-\mathrm{1}} \:-\:\mathrm{2}\:\mathrm{H}_{\boldsymbol{\mathrm{n}}-\mathrm{1}} \right)} \\ $$

Question Number 161130    Answers: 1   Comments: 0

Given P(x) is polynomial such that P(3x)= P β€²(x).P β€²β€²(x) . Find the tangent of curve y = P(x) parallel to the line y= 4xβˆ’2.

$$\:{Given}\:{P}\left({x}\right)\:{is}\:{polynomial}\:{such}\:{that} \\ $$$$\:{P}\left(\mathrm{3}{x}\right)=\:{P}\:'\left({x}\right).{P}\:''\left({x}\right)\:.\:{Find}\:{the}\:{tangent} \\ $$$$\:{of}\:{curve}\:{y}\:=\:{P}\left({x}\right)\:{parallel}\:{to}\:{the}\:{line} \\ $$$$\:{y}=\:\mathrm{4}{x}βˆ’\mathrm{2}.\: \\ $$

Question Number 161126    Answers: 1   Comments: 2

Question Number 161111    Answers: 1   Comments: 1

lim_(xβ†’0) ((x^2 +2cos xβˆ’2)/x^4 ) = (1/a) a=?

$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} +\mathrm{2cos}\:{x}βˆ’\mathrm{2}}{{x}^{\mathrm{4}} }\:=\:\frac{\mathrm{1}}{{a}} \\ $$$$\:\:\:\:{a}=? \\ $$

Question Number 161105    Answers: 1   Comments: 1

Question Number 161102    Answers: 1   Comments: 1

Question Number 161101    Answers: 1   Comments: 0

solve: ∫((x+1)/(x^2 βˆ’7xβˆ’3))dx

$${solve}: \\ $$$$\:\:\:\int\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} βˆ’\mathrm{7}{x}βˆ’\mathrm{3}}{dx} \\ $$

Question Number 161100    Answers: 0   Comments: 0

f(x^2 )= 2+∫_( 0) ^( x^2 ) f(y) (1βˆ’tan y)dy , βˆ€x∈R f(βˆ’Ο€)=?

$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{2}+\int_{\:\mathrm{0}} ^{\:\mathrm{x}^{\mathrm{2}} } \mathrm{f}\left(\mathrm{y}\right)\:\left(\mathrm{1}βˆ’\mathrm{tan}\:\mathrm{y}\right)\mathrm{dy}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{f}\left(βˆ’\pi\right)=? \\ $$

Question Number 161096    Answers: 0   Comments: 0

if x;y;z>0 and a;b;c>0 different in pairs and n;k∈N^βˆ— ((log x^n )/(b^k - c^k )) = ((log y^n )/(c^k - a^k )) = ((log z^n )/(a^k - b^k )) then find (√(xyz))

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{different}\:\mathrm{in}\:\mathrm{pairs}\:\mathrm{and}\:\:\mathrm{n};\mathrm{k}\in\mathbb{N}^{\ast} \\ $$$$\frac{\mathrm{log}\:\mathrm{x}^{\boldsymbol{\mathrm{n}}} }{\mathrm{b}^{\boldsymbol{\mathrm{k}}} \:-\:\mathrm{c}^{\boldsymbol{\mathrm{k}}} }\:=\:\frac{\mathrm{log}\:\mathrm{y}^{\boldsymbol{\mathrm{n}}} }{\mathrm{c}^{\boldsymbol{\mathrm{k}}} \:-\:\mathrm{a}^{\boldsymbol{\mathrm{k}}} }\:=\:\frac{\mathrm{log}\:\mathrm{z}^{\boldsymbol{\mathrm{n}}} }{\mathrm{a}^{\boldsymbol{\mathrm{k}}} \:-\:\mathrm{b}^{\boldsymbol{\mathrm{k}}} } \\ $$$$\mathrm{then}\:\mathrm{find}\:\:\sqrt{\boldsymbol{\mathrm{xyz}}} \\ $$

Question Number 161091    Answers: 1   Comments: 0

Solve for real numbers: (√(1 - x)) = 2x^2 - 1 - 2x (√(1 - x^2 ))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt{\mathrm{1}\:-\:\mathrm{x}}\:=\:\mathrm{2x}^{\mathrm{2}} \:-\:\mathrm{1}\:-\:\mathrm{2x}\:\sqrt{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 161114    Answers: 0   Comments: 0

Let f(x)= sin^3 (2x) for βˆ’(Ο€/4)≀x≀(Ο€/4) then Df^(βˆ’1) ((1/8))=(a/(b(√b))) so { ((a=?)),((b=?)) :}

$$\:\:{Let}\:{f}\left({x}\right)=\:\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{2}{x}\right)\:{for}\:βˆ’\frac{\pi}{\mathrm{4}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{4}} \\ $$$$\:{then}\:{Df}^{βˆ’\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{8}}\right)=\frac{{a}}{{b}\sqrt{{b}}}\:{so}\:\begin{cases}{{a}=?}\\{{b}=?}\end{cases} \\ $$

Question Number 161089    Answers: 3   Comments: 0

prove that I= ∫_0 ^( (Ο€/2)) ln ( 1+ sin (2 Ξ± )) dΞ± = 2G βˆ’ Ο€ ln ((√2) ) G: catalan constant

$$ \\ $$$$\:\:{prove}\:{that} \\ $$$$\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:\left(\:\mathrm{1}+\:{sin}\:\left(\mathrm{2}\:\alpha\:\right)\right)\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{2G}\:βˆ’\:\pi\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$

Question Number 161084    Answers: 0   Comments: 0

Question Number 161079    Answers: 1   Comments: 3

Question Number 161076    Answers: 1   Comments: 0

Ξ© = ∫_0 ^( ∞) ((ln (1+ x ))/((1+ x^( 2) )^( 2) )) dx = ? βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\:\left(\mathrm{1}+\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 161075    Answers: 0   Comments: 0

simplify Ξ£_(n=1) ^∞ (( n)/(( n^( 2) βˆ’(( 1)/4) )^( 3) )) = ?

$$ \\ $$$$\:\:\:\:{simplify} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{n}}{\left(\:{n}^{\:\mathrm{2}} βˆ’\frac{\:\mathrm{1}}{\mathrm{4}}\:\right)^{\:\mathrm{3}} }\:=\:? \\ $$$$ \\ $$

Question Number 161071    Answers: 2   Comments: 0

For a,b,c > 0 . Find (x,y,z) that satisfy this equation system ax + by = (xβˆ’y)^2 by + cz = (yβˆ’z)^2 cz + ax = (zβˆ’x)^2

$${For}\:\:{a},{b},{c}\:>\:\mathrm{0}\:. \\ $$$${Find}\:\:\left({x},{y},{z}\right)\:\:{that}\:\:{satisfy}\:\:{this}\:\:{equation}\:\:{system}\: \\ $$$$\:\:\:{ax}\:+\:{by}\:=\:\left({x}βˆ’{y}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:{by}\:+\:{cz}\:=\:\left({y}βˆ’{z}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:{cz}\:+\:{ax}\:=\:\left({z}βˆ’{x}\right)^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 161068    Answers: 2   Comments: 0

∫ ((2x)/((1βˆ’x^2 )(√(x^4 βˆ’1)))) dx =?

$$\:\:\:\:\:\int\:\frac{\mathrm{2}{x}}{\left(\mathrm{1}βˆ’{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{4}} βˆ’\mathrm{1}}}\:{dx}\:=? \\ $$

Question Number 161066    Answers: 2   Comments: 0

x_1 ,x_2 be the roots of the equation x^2 +x+m=0 & x_1 ^5 +x_2 ^5 = 2021. Find the sum of the possible values of m.

$$\:{x}_{\mathrm{1}} \:,{x}_{\mathrm{2}} \:{be}\:{the}\:{roots}\:{of}\:{the}\:{equation}\: \\ $$$$\:\:\:\:\:\:{x}^{\mathrm{2}} +{x}+{m}=\mathrm{0}\:\&\:{x}_{\mathrm{1}} ^{\mathrm{5}} +{x}_{\mathrm{2}} ^{\mathrm{5}} \:=\:\mathrm{2021}. \\ $$$$\:{Find}\:{the}\:{sum}\:{of}\:{the}\:{possible}\:{values} \\ $$$$\:\:{of}\:{m}. \\ $$

Question Number 161065    Answers: 1   Comments: 0

{ ((((x+abc))^(1/4) +((xβˆ’abc))^(1/8) = a)),((((x+abc))^(1/4) βˆ’((xβˆ’abc))^(1/8) = b)),((((x+abc))^(1/4) βˆ’((xβˆ’abc))^(1/4) = c)) :} find (√(x+abc)) +(√(xβˆ’abc))

$$\:\begin{cases}{\sqrt[{\mathrm{4}}]{{x}+{abc}}\:+\sqrt[{\mathrm{8}}]{{x}βˆ’{abc}}\:=\:{a}}\\{\sqrt[{\mathrm{4}}]{{x}+{abc}}\:βˆ’\sqrt[{\mathrm{8}}]{{x}βˆ’{abc}}\:=\:{b}}\\{\sqrt[{\mathrm{4}}]{{x}+{abc}}\:βˆ’\sqrt[{\mathrm{4}}]{{x}βˆ’{abc}}\:=\:{c}}\end{cases} \\ $$$$\:{find}\:\sqrt{{x}+{abc}}\:+\sqrt{{x}βˆ’{abc}} \\ $$

Question Number 161061    Answers: 1   Comments: 0

Question Number 161060    Answers: 1   Comments: 2

Given sin(5xβˆ’38)=cos(2x+16), 0°≀x≀90Β°, find the value of x

$$\mathrm{Given}\:\mathrm{sin}\left(\mathrm{5x}βˆ’\mathrm{38}\right)=\mathrm{cos}\left(\mathrm{2x}+\mathrm{16}\right),\:\mathrm{0}Β°\leqslant\mathrm{x}\leqslant\mathrm{90}Β°, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$

Question Number 161059    Answers: 0   Comments: 0

Find: 𝛀 =∫_( 0) ^( 1) ∫_( 0) ^( 1) (x^2 +2xy+x)ln(1 + (1/(x+y)))dxdy

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2xy}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right)\mathrm{dxdy} \\ $$

Question Number 161058    Answers: 1   Comments: 0

Solve the differential equation: x(y-1)dx + (x+1)dy = 0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$$\mathrm{x}\left(\mathrm{y}-\mathrm{1}\right)\mathrm{dx}\:+\:\left(\mathrm{x}+\mathrm{1}\right)\mathrm{dy}\:=\:\mathrm{0} \\ $$$$ \\ $$

Question Number 161039    Answers: 0   Comments: 0

let the differential equation: (1 + x) y^(β€²β€²) (x) + (1 - x) y^β€² (x) = ((1-x)/(1+x)) y(x) y(0) = 1 , y^β€² (0) = 0 then prove that: ∫_( 0) ^( ∞) (y^(β€²β€²) (x) + y^β€² (x) + y(x)) e^(-x) dx = (3/2)

$$\mathrm{let}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$$\left(\mathrm{1}\:+\:\mathrm{x}\right)\:\mathrm{y}^{''} \left(\mathrm{x}\right)\:+\:\left(\mathrm{1}\:-\:\mathrm{x}\right)\:\mathrm{y}^{'} \left(\mathrm{x}\right)\:=\:\frac{\mathrm{1}-\mathrm{x}}{\mathrm{1}+\mathrm{x}}\:\mathrm{y}\left(\mathrm{x}\right) \\ $$$$\mathrm{y}\left(\mathrm{0}\right)\:=\:\mathrm{1}\:,\:\mathrm{y}^{'} \left(\mathrm{0}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\left(\mathrm{y}^{''} \left(\mathrm{x}\right)\:+\:\mathrm{y}^{'} \left(\mathrm{x}\right)\:+\:\mathrm{y}\left(\mathrm{x}\right)\right)\:\mathrm{e}^{-\boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:=\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

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