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Question Number 162925 Answers: 1 Comments: 0

lim_(x→0) (1/x^4 )[(1+x+(x^2 /2)+(x^3 /3))^(1/(x+(x^2 /2)+(x^3 /3))) −(1+x+(x^2 /2)+(x^3 /3)+(x^4 /4))^(1/(x+(x^2 /2)+(x^3 /3)+(x^4 /4))) ]=?

Question Number 162924 Answers: 2 Comments: 0

𝛗 =∫_0 ^( ∞) (( e^( −x^( 2) ) .ln( x ))/( (√x))) dx=λ Γ((1/4)) λ=? ■

$$\: \\ $$$$\:\boldsymbol{\phi}\:=\int_{\mathrm{0}} ^{\:\infty} \frac{\:{e}^{\:−{x}^{\:\mathrm{2}} } .\mathrm{ln}\left(\:{x}\:\right)}{\:\sqrt{{x}}}\:{dx}=\lambda\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lambda=?\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare \\ $$$$ \\ $$

Question Number 162894 Answers: 1 Comments: 0

Question Number 162893 Answers: 2 Comments: 0

Ω=∫_0 ^( 1) ((( x^ )/(ln^ ( 1−x ))))^( 2) dx=^? ln ((( 27)/(16)) ) −−−−

$$ \\ $$$$\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\:{x}^{\:} }{\mathrm{ln}^{\:} \left(\:\mathrm{1}−{x}\:\right)}\right)^{\:\mathrm{2}} {dx}\overset{?} {=}\:\mathrm{ln}\:\left(\frac{\:\mathrm{27}}{\mathrm{16}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:−−−− \\ $$$$ \\ $$

Question Number 162877 Answers: 2 Comments: 0

Find: 𝛀 = ∫_( 0) ^( 1) (x^3 /(ln^2 (1 - x))) dx

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:\:=\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{ln}^{\mathrm{2}} \:\left(\mathrm{1}\:-\:\mathrm{x}\right)}\:\mathrm{dx} \\ $$

Question Number 162876 Answers: 0 Comments: 0

Prove that: ∫_( 0) ^( (𝛑/2)) (xcotx ∙ lncos^2 x + ln^2 cosx)dx = (π^3 /(24))

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\left(\mathrm{xcot}\boldsymbol{\mathrm{x}}\:\centerdot\:\mathrm{lncos}^{\mathrm{2}} \boldsymbol{\mathrm{x}}\:+\:\mathrm{ln}^{\mathrm{2}} \mathrm{cos}\boldsymbol{\mathrm{x}}\right)\mathrm{dx}\:=\:\frac{\pi^{\mathrm{3}} }{\mathrm{24}} \\ $$

Question Number 162872 Answers: 1 Comments: 1

Question Number 162866 Answers: 1 Comments: 0

Question Number 162865 Answers: 0 Comments: 4

Question Number 162864 Answers: 2 Comments: 0

Question Number 162860 Answers: 2 Comments: 0

Question Number 162859 Answers: 0 Comments: 0

Prove that: ∫_( 0) ^( (𝛑/2)) ((e^(cos 2x) ∙ sin(x + sin 2x))/(sin x)) dx = ((πe)/2)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\frac{\mathrm{e}^{\boldsymbol{\mathrm{cos}}\:\mathrm{2}\boldsymbol{\mathrm{x}}} \:\centerdot\:\mathrm{sin}\left(\mathrm{x}\:+\:\mathrm{sin}\:\mathrm{2x}\right)}{\mathrm{sin}\:\mathrm{x}}\:\mathrm{dx}\:=\:\frac{\pi{e}}{\mathrm{2}} \\ $$

Question Number 162856 Answers: 0 Comments: 0

Question Number 162854 Answers: 0 Comments: 0

let a;b;c≥0 and a+b+c=3 prove that: ((a - 1)/( (√(b + 3)))) + ((b - 1)/( (√(c + 3)))) + ((c - 1)/( (√(a + 3)))) ≥ 0

$$\mathrm{let}\:\:\mathrm{a};\mathrm{b};\mathrm{c}\geqslant\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{3}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{a}\:-\:\mathrm{1}}{\:\sqrt{\mathrm{b}\:+\:\mathrm{3}}}\:+\:\frac{\mathrm{b}\:-\:\mathrm{1}}{\:\sqrt{\mathrm{c}\:+\:\mathrm{3}}}\:+\:\frac{\mathrm{c}\:-\:\mathrm{1}}{\:\sqrt{\mathrm{a}\:+\:\mathrm{3}}}\:\geqslant\:\mathrm{0} \\ $$

Question Number 162847 Answers: 2 Comments: 0

Question Number 162845 Answers: 1 Comments: 1

Question Number 162834 Answers: 5 Comments: 0

Question Number 162833 Answers: 1 Comments: 0

Question Number 162827 Answers: 1 Comments: 0

Question Number 162825 Answers: 2 Comments: 0

Question Number 162823 Answers: 2 Comments: 0

Given: x.p(x−1)=(x−5).p(x) and p(−1)=1. Find p((1/2)).

$$\:\:{Given}:\:\:{x}.{p}\left({x}−\mathrm{1}\right)=\left({x}−\mathrm{5}\right).{p}\left({x}\right) \\ $$$$\:\:{and}\:{p}\left(−\mathrm{1}\right)=\mathrm{1}.\: \\ $$$$\:\:{Find}\:{p}\left(\frac{\mathrm{1}}{\mathrm{2}}\right). \\ $$

Question Number 163727 Answers: 0 Comments: 0

Question Number 163724 Answers: 0 Comments: 0

Question Number 162819 Answers: 0 Comments: 1

#combinatorial mathematics# In how many subsets of 10 members of the set , { 1, 2, ..., 20 } is there no difference between two members of 5 ? a: 2^( 10) b : 3^( 5) c : 2^( 8) d: 10^( 4)

$$ \\ $$$$\:\:\:\:\:#{combinatorial}\:{mathematics}# \\ $$$$\:\:\:\:\:\:\mathrm{I}{n}\:{how}\:{many}\:\:{subsets}\:{of}\:\mathrm{10}\: \\ $$$${members}\:{of}\:\:{the}\:{set}\:,\:\left\{\:\mathrm{1},\:\mathrm{2},\:...,\:\mathrm{20}\:\right\} \\ $$$$\:\:\:{is}\:{there}\:{no}\:{difference}\:{between} \\ $$$$\:{two}\:{members}\:{of}\:\:\:\:\mathrm{5}\:?\: \\ $$$${a}:\:\:\mathrm{2}^{\:\mathrm{10}} \:\:\:\:\:\:{b}\::\:\:\mathrm{3}^{\:\mathrm{5}} \:\:\:\:\:\:\:{c}\::\:\:\:\mathrm{2}^{\:\mathrm{8}} \:\:\:\:\:\:\:\:{d}:\:\:\:\mathrm{10}^{\:\mathrm{4}} \\ $$

Question Number 162814 Answers: 2 Comments: 0

Question Number 162811 Answers: 1 Comments: 0

I = ∫_0 ^( ∞) (( tan^( −1) (x ))/(( 1+x^( 2) )^( 2) )) dx = ? −−−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \:\left({x}\:\right)}{\left(\:\mathrm{1}+{x}^{\:\mathrm{2}} \:\right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:\:−−−−−−−−−− \\ $$

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