[THOUGHT] We need to compute the limit:

Let's denote:

We need to expand F(x) as a power series around x=0 up to order x^3, because the limit is of the form F(x)/x^3 → 2 as x→0. That means F(x) must behave like 2x^3 + o(x^3) near 0. So we need to find a and b such that the coefficients of x^0, x^1, x^2 vanish, and the coefficient of x^3 equals 2.

First, compute the integral:

But we can also expand it as a series. Alternatively, we can expand the integrand as a geometric series:

Integrate termwise:

Thus,

Now,

Next, expand bx cos x:

So,

Now sum:

Combine like terms: Coefficient of x: $\frac{a}{2} + b$. Coefficient of x^3: $\frac{a}{6} - \frac{b}{2}$.

For the limit to exist (finite), the coefficient of x must be zero, otherwise F(x) ~ C x, then F(x)/x^3 ~ C/x^2 → ∞ (unless C=0). So we need:

Then the coefficient of x^3 becomes the leading term. The limit is then:

We are given that this limit equals 2:

Solve (1) and (2). From (1): b = -a/2. Substitute into (2):

Thus a = 24/5 = 4.8. Then b = -a/2 = -12/5 = -2.4. Now compute (a+b)^2: a+b = 24/5 - 12/5 = 12/5 = 2.4. So (a+b)^2 = (12/5)^2 = 144/25. Then 100(a+b)^2 = 100 * 144/25 = (100/25)144 = 4144 = 576. Thus the answer is 576.

But we should verify with Python to ensure no algebraic mistakes. We'll compute the limit directly using sympy to confirm. [/THOUGHT] [PYTHON] import sympy as sp

Define symbols

x, a, b = sp.symbols('x a b')

Define the integral

integral = sp.integrate(1/(1 - t**2), (t, 0, x))

Define F(x)

F = (a/2) * integral + b * x * sp.cos(x)

Compute limit of F(x)/x^3 as x->0

limit_expr = sp.limit(F / x**3, x, 0) print("Limit expression (symbolic):", limit_expr)

The limit should be an expression in a and b.

Set up equations: coefficient of x^0? Actually limit_expr is a rational expression.

Let's simplify.

limit_simplified = sp.simplify(limit_expr) print("Simplified limit:", limit_simplified)

We know limit = 2.

Solve for a and b.

But we also need condition that lower order terms vanish: i.e., the limit exists.

Actually, sympy's limit will give the value if the limit exists.

Let's compute the series expansion of F(x) up to x^3 to see conditions.

series_F = sp.series(F, x, 0, 4).removeO() print("Series expansion of F(x) up to x^3:", series_F)

The series expansion gives terms: coefficient of x^1 and x^3.

For limit to exist, coefficient of x^1 must be zero.

coeff_x = sp.series(F, x, 0, 2).removeO().coeff(x, 1) print("Coefficient of x:", coeff_x)

Solve coeff_x = 0 and limit_simplified = 2.

sol = sp.solve([coeff_x, limit_simplified - 2], [a, b]) print("Solutions for a, b:", sol)

Compute (a+b)^2 and then 100*(a+b)^2

if sol: a_val = sol[a] b_val = sol[b] expr = 100 * (a_val + b_val)*2 print("100(a+b)^2 =", expr) # Evaluate to integer print("Integer value:", expr.evalf()) [/PYTHON]

[THOUGHT] The Python output shows an error: "NotImplementedError: solving Abs(a + 2*b) when the argument is not real or imaginary." This suggests that the symbolic solving encountered an absolute value due to the limit expression. However, my earlier manual derivation gave a clear result. Since the Python code had issues, I should trust my manual calculation which was verified step-by-step and gave 576. The error is likely due to sympy's handling of the limit expression, but the manual derivation is straightforward and correct. Therefore, the final answer is 576.

[/THOUGHT] $\boxed{576}$