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Question Number 154225 Answers: 1 Comments: 1

In the expansion (1+(1/2)x)^n , the coefficient of x^m is (5/4) times the coefficient of x^(m+1) . a) Show that 5n−13m=8 b) If m and n are positive integers, such that m≤n, determine the smallest values of m and n.

$$\mathrm{In}\:\mathrm{the}\:\mathrm{expansion}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}\right)^{{n}} ,\:\mathrm{the}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:{x}^{{m}} \:\mathrm{is}\:\frac{\mathrm{5}}{\mathrm{4}}\:\mathrm{times}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{m}+\mathrm{1}} . \\ $$$$\left.{a}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{5}{n}−\mathrm{13}{m}=\mathrm{8} \\ $$$$\left.{b}\right)\:\mathrm{If}\:{m}\:\mathrm{and}\:{n}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:{m}\leqslant{n},\:\mathrm{determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{values}\:\mathrm{of} \\ $$$$\:\:\:\:\:{m}\:\mathrm{and}\:{n}. \\ $$

Question Number 154216 Answers: 3 Comments: 0

Solve the equation for reals ((cos^2 x−(1/5)sin^2 x+1)/(sin^2 x−(1/5)cos^2 x+1)) = 5tan^2 x

$${Solve}\:{the}\:{equation}\:{for}\:{reals}\: \\ $$$$\:\frac{\mathrm{cos}\:^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{5}}\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} {x}−\frac{\mathrm{1}}{\mathrm{5}}\mathrm{cos}\:^{\mathrm{2}} {x}+\mathrm{1}}\:=\:\mathrm{5tan}\:^{\mathrm{2}} {x}\: \\ $$

Question Number 154208 Answers: 0 Comments: 0

∵∴∵∴prove that ∫_0 ^∞ ∫_0 ^∞ ((log(1−e^(−x) )(yLi_2 (e^(−x−y) )+Li_3 (e^(−x−y) ))/(1−e^(x+y) ))e^(x+y) dxdy=((21)/8)ζ(6)+ζ^2 (3) ∵∴∵by MATH.AMIN∴∵∴

$$\because\therefore\because\therefore{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{{log}\left(\mathrm{1}−{e}^{−{x}} \right)\left({yLi}_{\mathrm{2}} \left({e}^{−{x}−{y}} \right)+{Li}_{\mathrm{3}} \left({e}^{−{x}−{y}} \right)\right.}{\mathrm{1}−{e}^{{x}+{y}} }{e}^{{x}+{y}} {dxdy}=\frac{\mathrm{21}}{\mathrm{8}}\zeta\left(\mathrm{6}\right)+\zeta^{\mathrm{2}} \left(\mathrm{3}\right) \\ $$$$\because\therefore\because\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{MATH}}.\boldsymbol{\mathrm{AMIN}}\therefore\because\therefore \\ $$

Question Number 154202 Answers: 1 Comments: 0

𝛀 =∫_( 0) ^( ∞) ((ln∙(1 + x))/(x∙(x^2 + x + 1))) dx = ?

$$\boldsymbol{\Omega}\:\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{ln}\centerdot\left(\mathrm{1}\:+\:\mathrm{x}\right)}{\mathrm{x}\centerdot\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)}\:\mathrm{dx}\:=\:? \\ $$

Question Number 154194 Answers: 0 Comments: 1

Question Number 154195 Answers: 3 Comments: 0

∫((x+sinx)/(1+cosx))dx please,help me

$$\int\frac{{x}+{sinx}}{\mathrm{1}+{cosx}}{dx}\:\:\:\: \\ $$$${please},{help}\:{me} \\ $$

Question Number 154192 Answers: 1 Comments: 0

Ω :=∫_0 ^( 1) (( x.sin(ln(x)))/(1−x))dx method 1 Ω= Im[∫_0 ^( 1) (( x^( i+1) )/(1−x)) dx=Φ] Φ = ∫_0 ^( 1) (( x^( i+1) +x^( i+2) )/(1−x^( 2) ))dx =^(x^( 2) =t) (1/2)∫_0 ^( 1) (( t^( (i/2)) −t^((i+1)/2) )/(1−t))dt = (1/2) { ψ (1 +((i+1)/2))−ψ (1+(i/2))} = (1/2) { (2/(1+i)) + ψ ( ((1+i)/2) )−(2/i)−ψ ((i/2))} = ((−i)/(−1+i)) + (1/2) {ψ (((1+i)/2))−ψ((i/2) )} = −(1/2) +(i/(2 )) +(1/2) {ψ(((1+i)/2))−ψ((i/2))} Ω = Im (Φ )= (1/2) +(1/2) Im(ψ((1/2) +(i/2)))−(1/2) Im((i/2)) = (1/(2 )) { 1 +(π/2) tanh((π/2)) −1−(π/2) coth((π/2))} =(π/4) {((−1)/(sinh((π/2)).cosh((π/2))))}=((−π)/(2 sinh (π ))) ✓

Question Number 154186 Answers: 1 Comments: 0

Question Number 154177 Answers: 1 Comments: 2

Question Number 154175 Answers: 2 Comments: 1

Question Number 154172 Answers: 0 Comments: 1

Question Number 154148 Answers: 1 Comments: 0

z^4 - ((50)/(2z^4 - 7)) = 14 ⇒ z = ?

$$\boldsymbol{{z}}^{\mathrm{4}} \:-\:\frac{\mathrm{50}}{\mathrm{2}\boldsymbol{{z}}^{\mathrm{4}} \:-\:\mathrm{7}}\:=\:\mathrm{14}\:\:\:\Rightarrow\:\:\:\boldsymbol{{z}}\:=\:? \\ $$

Question Number 154143 Answers: 2 Comments: 0

Question Number 154142 Answers: 2 Comments: 0

Question Number 154138 Answers: 1 Comments: 0

Question Number 154133 Answers: 2 Comments: 0

Given f(x)=((x+(√(1+x^2 ))))^(1/3) +((x−(√(1+x^2 ))))^(1/3) Find f^(−1) (x)=?

$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\sqrt[{\mathrm{3}}]{\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}+\sqrt[{\mathrm{3}}]{\mathrm{x}−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }} \\ $$$$\mathrm{Find}\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)=? \\ $$

Question Number 154131 Answers: 0 Comments: 1

A long distance runner runs 14km 30°north of east, 7km 60° north of west, 6km 30° south of west, and finally 4km south. Find his final distance and direction relative to the starting point?

$$ \\ $$A long distance runner runs 14km 30°north of east, 7km 60° north of west, 6km 30° south of west, and finally 4km south. Find his final distance and direction relative to the starting point?

Question Number 154200 Answers: 1 Comments: 0

g(((x−1)/(x+1)))=((7x+3)/(x+1)) and f(x^2 −2x+3)=3x^2 −6x+7 find (f+g)(x)=?

$${g}\left(\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\right)=\frac{\mathrm{7}{x}+\mathrm{3}}{{x}+\mathrm{1}}\:\:{and}\:\:{f}\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{3}\right)=\mathrm{3}{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{7} \\ $$$${find}\:\:\left({f}+{g}\right)\left({x}\right)=?\:\:\: \\ $$

Question Number 154116 Answers: 0 Comments: 0