To solve the problem, we need to find the value of the integral \( \int_{-4}^{4} f(x^2) \, dx \) given the equations:
1. \( f(x^2) + g(4 - x) = 4x^3 \)
2. \( g(4 - x) + g(x) = 0 \)
### Step 1: Integrate the first equation from 0 to 4
We start by integrating the first equation from 0 to 4:
\[
\int_{0}^{4} \left( f(x^2) + g(4 - x) \right) \, dx = \int_{0}^{4} 4x^3 \, dx
\]
### Step 2: Evaluate the right-hand side
Calculating the right-hand side:
\[
\int_{0}^{4} 4x^3 \, dx = 4 \left[ \frac{x^4}{4} \right]_{0}^{4} = \left[ x^4 \right]_{0}^{4} = 4^4 - 0^4 = 256
\]
### Step 3: Change of variable for the left-hand side
Now we focus on the left-hand side. We can change the variable in the second term of the left-hand side:
Let \( u = 4 - x \), then \( du = -dx \). When \( x = 0 \), \( u = 4 \) and when \( x = 4 \), \( u = 0 \).
Thus,
\[
\int_{0}^{4} g(4 - x) \, dx = \int_{4}^{0} g(u) (-du) = \int_{0}^{4} g(u) \, du = \int_{0}^{4} g(x) \, dx
\]
### Step 4: Combine the integrals
Now we can rewrite the left-hand side:
\[
\int_{0}^{4} f(x^2) \, dx + \int_{0}^{4} g(x) \, dx = 256
\]
### Step 5: Use the second equation
From the second equation \( g(4 - x) + g(x) = 0 \), we have:
\[
g(4 - x) = -g(x)
\]
This implies that \( g(x) \) is an odd function. Therefore, the integral of \( g(x) \) from -4 to 4 is zero:
\[
\int_{-4}^{4} g(x) \, dx = 0
\]
### Step 6: Relate the integrals
Now, we can relate the integrals:
\[
\int_{0}^{4} g(x) \, dx = -\int_{0}^{4} g(4 - x) \, dx
\]
This means that:
\[
\int_{0}^{4} g(x) \, dx = 0
\]
### Step 7: Substitute back into the equation
Substituting this back into our equation gives:
\[
\int_{0}^{4} f(x^2) \, dx + 0 = 256
\]
Thus:
\[
\int_{0}^{4} f(x^2) \, dx = 256
\]
### Step 8: Find the value of the integral from -4 to 4
Since \( f(x^2) \) is an even function (because \( f(x^2) \) depends only on \( x^2 \)), we have:
\[
\int_{-4}^{4} f(x^2) \, dx = 2 \int_{0}^{4} f(x^2) \, dx = 2 \times 256 = 512
\]
### Final Answer
Thus, the value of \( \int_{-4}^{4} f(x^2) \, dx \) is \( \boxed{512} \).
