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 #2
avatar+26396 
+1

Let

n

and

k

be positive integers such that (nk):(nk+1)=4:11. Find the smallest possible value of .

 

(nk)(nk+1)=411|(nk+1)=(nk)nkk+1(nk)(nk)nkk+1=411k+1nk=41111(k+1)=4(nk)11k+11=4n4k4n15k=114n=11+15kn=11+15k4=11+11+15k+kk4=12+16k(1+k)4=12+16k4(1+k)4=3+4k(1+k)4=an=3+4kaa=1+k44a=1+kk=4a1,aZn=3+4ka|k=4a1=3+4(4a1)a=3+16a4an=1+15anmin=1+151|a=1nmin=14

 

k=4a1|a=1k=411k=3(143)(144)=411

 

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Mar 26, 2019
 #7
avatar+26396 
+1

2. 

∫5 3 (1/(x^2-6x+13))dx

531x26x+13 dx

 

531x26x+13 dx=531(x3)29+13 dx=531(x3)2+4 dx=5314(1+(x3)24) dx=53121+(x32)2 dx=125311+(x32)2 dxsubstitute: x32=sinh(θ), 12dx=cosh(θ)dθ, dx=2cosh(θ)dθx=3θ=sinh1(332)=0,x=5θ=sinh1(532)=sinh1(1)=ln(1+12+1)=0.88137358702=120.88137358702011+(sinh(θ))22cosh(θ) dθ=0.881373587020cosh(θ)cosh(θ) dθ=0.881373587020dθ=[θ]0.881373587020=0.88137358702531x26x+13 dx=sinh1(1)=0.88137358702

 

 

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Mar 26, 2019
 #6
avatar+26396 
+2

1. 

∫1/6 0 (1/(sqrt(1-9x^2))dx

160119x2dx

 

160119x2 dx=16011(3x)2 dxsubstitute: u=3x, du=3 dx,x=0u=30=0,x=16u=316=12=12011u2du3=1312011u2 dusubstitute: u=sin(θ), du=cos(θ)dθ,u=0θ=arcsin(0)=0,u=12θ=arcsin(12)=π6=13π60cos(θ)1(sin(θ))2 dθ=13π60cos(θ)cos(θ) dθ=13π60dθ=13[θ]π60=13π6=π18

 

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Mar 26, 2019

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