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Given-F-x-1-2-x-1-x-1-f-t-dt-Show-that-F-is-defined-continuous-and-derivable-And-find-its-derivative-

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Question Number 99261 by Ar Brandon last updated on 19/Jun/20
Given F(x)=(1/2)∫((x+1)/(x−1))f(t)dt Show that F is defined, continuous, and derivable. And find its derivative
GivenF(x)=12∫x+1x−1f(t)dtShowthatFisdefined,continuous,andderivable.Andfinditsderivative
Answered by abdomathmax last updated on 19/Jun/20
F(x) =(1/2)((x+1)/(x−1)) ∫^x f(t)dt ⇒F^′ (x)=(1/2)(((x+1)/(x−1)))^′ ∫^x f(t)dt +((x+1)/(2(x−1)))f^′ (x) =(1/2)×((x−1−(x+1))/((x−1)^2 )) ∫^x f(t)dt +((x+1)/(2(x−1)))f^′ (x) =−(1/((x−1)^2 )) ∫^x f(t)dt+((x+1)/(2(x−1)))f^′ (x)
F(x)=12x+1x−1∫xf(t)dt⇒F′(x)=12(x+1x−1)′∫xf(t)dt+x+12(x−1)f′(x)=12×x−1−(x+1)(x−1)2∫xf(t)dt+x+12(x−1)f′(x)=−1(x−1)2∫xf(t)dt+x+12(x−1)f′(x)

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