Question Number 104383 by abony1303 last updated on 21/Jul/20

When f(x) is a differentiable function  satisfying  x∙f(x)=x^2 +∫_0 ^( x) (x−t)∙f ′(t)dt  Find ⇒ f(1)

Commented byabony1303 last updated on 21/Jul/20

Pls help

Answered by mathmax by abdo last updated on 21/Jul/20

xf(x)=x^2  +∫_0 ^x (x−t)f^′ (t)dt   x=1 ⇒f(1) =1 +∫_0 ^1 (1−t)f^′ (t)dt  by parts  ∫_0 ^1 (1−t)f^′ (t)dt =[(1−t)f(t)]_0 ^1 −∫_0 ^1 (−1)f(t)dt  =f(0)+∫_0 ^1 f(t)dt  but f(0)=0 ⇒∫_0 ^1 (1−t)f^′ (t)dt =∫_0 ^1 f(t)dt ⇒  f(1) =1+∫_0 ^1 f(t)dt