Let f x be a differentiable function defined on 0 , 2 such that f ' x = f ' 2 - x for all x ∈ 0 , 2 , f 0 = 1 and f 2 = e 2 . Then the value of ∫ 0 2 f x d x is

Question

Let f x be a differentiable function defined on 0 , 2 such that f ' x = f ' 2 - x for all x ∈ 0 , 2 , f 0 = 1 and f 2 = e 2 . Then the value of ∫ 0 2 f x d x is, 2 1 + e 2, 1 + e 2, 1 - e 2, 2 1 - e 2

MARKS App · Accepted Answer

f ' x = f ' 2 - x f x = - f 2 - x + c put x = 0 f 0 = - f 2 + c c = f 0 + f 2 = 1 + e 2 so, f x + f 2 - x = 1 + e 2 I = ∫ 0 2 f x d x I = ∫ 0 2 f 2 - x d x 2 I = ∫ 0 2 f x + f 2 - x d x 2 I = 1 + e 2 ∫ 0 2 d x I = 1 + e 2

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