Let-f-x-be-a-function-such-that-f-1-x-x-3-f-x-0-What-is-1-1-f-x-dx-equal-to-a-2f-1-b-0-c-2f-1-d-4f-1-

Question Number 12302 by Gaurav3651 last updated on 18/Apr/17

L e t f (x) b e a f u n c t i o n s u c h t h a t

f^{'} (1 / x) + x^{3} f^{'} (x) = 0 .

W h a t i s \underset{- 1}{\int^{1}} f (x) d x e q u a l t o ?

(a) 2 f (1)

(b) 0

(c) 2 f (- 1)

(d) 4 f (1)

Answered by mrW1 last updated on 18/Apr/17

f^{'} (1 / x) = - x^{3} f^{'} (x)

- \frac{1}{x^{2}} f^{'} (\frac{1}{x}) = {x f}^{'} (x)

\int [- \frac{1}{x^{2}} f^{'} (\frac{1}{x})] d x = \int {x f}^{'} (x) d x

\int f^{'} (\frac{1}{x}) d (\frac{1}{x}) = \int x d [f (x)] + C

f (\frac{1}{x}) = x f (x) - \int f (x) d x + C

\Rightarrow \int f (x) d x = x f (x) - f (\frac{1}{x}) + C

\int_{- 1}^{1} f (x) d x = {[x f (x) - f (\frac{1}{x})]}_{- 1}^{1}

= [1 \times f (1) - f (1)] - [- 1 \times f (- 1) - f (- 1)]

= 2 f (- 1)

\Rightarrow A n s w e r (c)

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