Let ∫ 1 - ln ⁡ x x 2 d x = f x , for all positive x . If f e = 1 e , then f 2 + f 4 is equal to

Question

Let ∫ 1 - ln ⁡ x x 2 d x = f x , for all positive x . If f e = 1 e , then f 2 + f 4 is equal to, ln ⁡ 4, ln ⁡ 2, 1, 0

MARKS App · Accepted Answer

f x = ∫ 1 x 2 d x - ∫ ln ⁡ x ⋅ 1 x 2 d x = - 1 x - ln ⁡ x - 1 x + ∫ 1 x 2 d x = - 1 x + ln ⁡ x x + 1 x + c = ln ⁡ x x + c Now, f e = 1 e ⇒ c = 0 Hence, f 2 + f 4 = ln ⁡ 2 2 + ln ⁡ 4 4 = ln 2

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