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Question Number 135784 by benjo_mathlover last updated on 16/Mar/21

Given  { ((f(3)=4 , f ′(3)=−2)),((f(8)=5 , f ′(8)=3)) :}  find ∫_3 ^( 8)  x f ′′(x) dx .

$${Given}\:\begin{cases}{{f}\left(\mathrm{3}\right)=\mathrm{4}\:,\:{f}\:'\left(\mathrm{3}\right)=−\mathrm{2}}\\{{f}\left(\mathrm{8}\right)=\mathrm{5}\:,\:{f}\:'\left(\mathrm{8}\right)=\mathrm{3}}\end{cases} \\ $$$${find}\:\int_{\mathrm{3}} ^{\:\mathrm{8}} \:{x}\:{f}\:''\left({x}\right)\:{dx}\:. \\ $$

Answered by Ar Brandon last updated on 16/Mar/21

∫_3 ^8 xf′′(x)dx={xf ′(x)−∫f ′(x)dx}_3 ^8                             ={xf ′(x)−f(x)}_3 ^8 ={8(3)−5}−{3(−2)−4}=29

$$\int_{\mathrm{3}} ^{\mathrm{8}} \mathrm{xf}''\left(\mathrm{x}\right)\mathrm{dx}=\left\{\mathrm{xf}\:'\left(\mathrm{x}\right)−\int\mathrm{f}\:'\left(\mathrm{x}\right)\mathrm{dx}\right\}_{\mathrm{3}} ^{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left\{\mathrm{xf}\:'\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{x}\right)\right\}_{\mathrm{3}} ^{\mathrm{8}} =\left\{\mathrm{8}\left(\mathrm{3}\right)−\mathrm{5}\right\}−\left\{\mathrm{3}\left(−\mathrm{2}\right)−\mathrm{4}\right\}=\mathrm{29} \\ $$

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