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Question: Let ƒ(x) and g(x) be two functions satisfying ƒ(x<sup>2</sup>) + g(4 –x) = 4x<sup>3</sup>, g(4 – x)...

Let ƒ(x) and g(x) be two functions satisfying ƒ(x2) + g(4 –x)

= 4x3, g(4 – x) + g(x) = 0, then the value of ∫−44ƒ(x2)\int_{- 4}^{4}{ƒ(x^{2})}dx is-

A

512

B

64

C

256

D

0

Answer

512

Explanation

Solution

I = ∫−44f(x2)dx\int_{- 4}^{4}{f(x^{2})dx} = 2∫04f(x2)dx2\int_{0}^{4}{f(x^{2})dx}

= 2∫04{4x3−g(4−x)}dx2\int_{0}^{4}{\{ 4x^{3} - g(4 - x)\} dx}