   Chapter 7.P, Problem 11P

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# If 0 < a < b , find lim t → 0 { ∫ 0 1 [ b x + a ( 1 − x ) ] t d x } 1 / t

To determine

To Find: limt0{01[bx+a(1x)]tdx}1t;0<a<b

Explanation

Let bx+a(1x)=u,(ba)dx=du

When x=0,u=a

When x=1,u=b

limt0{01[bx+a(1x)]tdx}1t=limt0{1baabutdu}1t=1balimt0{abutdu}1t=1balimt0{[ut+1t+1]ab}1t=1balimt0{[bt+1at+1t+1]}1t

Letz={[bt+1at+1t+1]}1tlnz=1tln[bt+1at+1t+1]=1t

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