Let f(x) and g(x) be two functions satisfying `f(x^(2))+g(4-x)=4x^(3), g(4-x)+g(x)=0`, then the value of `int_(-4)^(4)f(x^(2))dx` is :

Options

A. 512
B. 64
C. 256
D. 0

Answer

To solve the given problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given equations
We have two equations:
1. \( f(x^2) + g(4 - x) = 4x^3 \)
2. \( g(4 - x) + g(x) = 0 \)

From the second equation, we can express \( g(4 - x) \) in terms of \( g(x) \):
\[ g(4 - x) = -g(x) \]

### Step 2: Substitute \( g(4 - x) \) in the first equation
Substituting \( g(4 - x) = -g(x) \) into the first equation gives:
\[ f(x^2) - g(x) = 4x^3 \]
Thus, we can rearrange this to:
\[ f(x^2) = 4x^3 + g(x) \]

### Step 3: Integrate both sides from 0 to 4
Now we will integrate both sides from \( x = 0 \) to \( x = 4 \):
\[ \int_0^4 f(x^2) \, dx = \int_0^4 (4x^3 + g(x)) \, dx \]

### Step 4: Calculate the right-hand side
We can split the integral on the right:
\[ \int_0^4 (4x^3 + g(x)) \, dx = \int_0^4 4x^3 \, dx + \int_0^4 g(x) \, dx \]

Calculating \( \int_0^4 4x^3 \, dx \):
\[ \int_0^4 4x^3 \, dx = 4 \left[ \frac{x^4}{4} \right]_0^4 = \left[ x^4 \right]_0^4 = 4^4 - 0 = 256 \]

Thus, we have:
\[ \int_0^4 f(x^2) \, dx = 256 + \int_0^4 g(x) \, dx \]

### Step 5: Use the property of \( g(x) \)
From the second equation \( g(4 - x) + g(x) = 0 \), we can derive:
\[ g(4 - x) = -g(x) \]

Using the property of integrals:
\[ \int_0^4 g(4 - x) \, dx = \int_0^4 -g(x) \, dx \]
This implies:
\[ \int_0^4 g(4 - x) \, dx = -\int_0^4 g(x) \, dx \]

Since \( g(4 - x) \) is equal to \( g(x) \) when integrated over the interval \( [0, 4] \), we can conclude:
\[ \int_0^4 g(x) \, dx = -\int_0^4 g(x) \, dx \implies 2\int_0^4 g(x) \, dx = 0 \implies \int_0^4 g(x) \, dx = 0 \]

### Step 6: Substitute back into the integral
Now substituting back, we have:
\[ \int_0^4 f(x^2) \, dx = 256 + 0 = 256 \]

### Step 7: Find the integral from -4 to 4
Since \( f(x^2) \) is an even function (as \( x^2 \) is even), we can use the property of even functions:
\[ \int_{-4}^{4} f(x^2) \, dx = 2 \int_0^4 f(x^2) \, dx = 2 \times 256 = 512 \]

### Final Answer
Thus, the value of \( \int_{-4}^{4} f(x^2) \, dx \) is \( \boxed{512} \).