[THOUGHT] I will expand the expression around x=0 using series (via sympy), then impose that the limit exists and equals 2, which will determine a and b. Then I’ll compute 100(a+b)^2. [/THOUGHT] [PYTHON] import sympy as sp

Define symbols

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

Define the integral term

t = sp.symbols('t') integrand = 1/(1 - t**2) I = sp.integrate(integrand, (t, 0, x))

Expression inside the limit

expr = (a/sp.Integer(2)) * I + bxsp.cos(x)

Series expansion of I around 0 up to x^5

I_series = sp.series(I, x, 0, 7) # more terms to be safe print("I series:", I_series)

Series expansion of expr up to x^5

expr_series = sp.series(expr, x, 0, 7) print("expr series:", expr_series)

Now divide by x^3 and expand

limit_expr = sp.series(expr / x**3, x, 0, 3) # up to x^0 term and maybe more print("limit expression series:", limit_expr)

Extract coefficients of x^-2, x^0 etc. from expr/x^3 directly via series expansion

full_series = sp.series(expr / x**3, x, 0, 5) print("Full series of expr/x^3:", full_series)

Let's get the simplified form and then series again to clearly see coefficients

simplified = sp.simplify(expr) limit_series = sp.series(simplified / x**3, x, 0, 5) print("Simplified expr/x^3 series:", limit_series)

Manually compute series for better control

I_series_explicit = sp.series(1 + t2 + t4 + t**6, t, 0, 7) # not needed; using sympy is fine

From the full_series, we determine conditions on a and b for limit to be finite and =2

Let's expand analytically with Rational series

I_series2 = sp.series(I, x, 0, 7).removeO() cos_series = sp.series(sp.cos(x), x, 0, 7).removeO() expr_series2 = (a/sp.Integer(2))I_series2 + bx*cos_series expr_div_x3_series = sp.series(expr_series2 / x**3, x, 0, 5).removeO() print("expr_div_x3_series (no O):", expr_div_x3_series)

Now solve for a and b requiring:

1) terms with negative powers of x vanish so limit is finite

2) constant term equals 2

Let's get series as polynomial in x (including negative powers) and inspect coefficients

expr_div = sp.simplify(expr_series2 / x**3) print("expr_div simplified:", expr_div)

Expand around 0 as series with negative powers explicitly

series_expr_div = sp.series(expr_div, x, 0, 5).removeO() print("Series of expr_div:", series_expr_div)

Extract coefficients for x^-2 (if any) and constant term

Write series_expr_div as sum c_k x^k

poly = sp.series(expr_div, x, 0, 5).removeO() print("poly:", poly)

Let's force it into a series expression by expanding and collecting

poly_expanded = sp.expand(poly) print("expanded poly:", poly_expanded)

Extract coefficients for x**(-2), x**0

coeff_x_minus2 = sp.residue(expr_div, x, 0) # not residue, better just use series

Instead parse powers manually

terms = sp.Add.make_args(poly_expanded) coeffs = {} for term in terms: term = sp.simplify(term) if term.is_Number: power = 0 coeffs[0] = coeffs.get(0, 0) + term else: # term like c*x^k c = term.as_coeff_exponent(x)[0] k = term.as_coeff_exponent(x)[1] coeffs[int(k)] = coeffs.get(int(k), 0) + c

print("coefficients by power:", coeffs)

Condition: coefficient at power -2 must be 0 (no 1/x^2 term)

And constant term (power 0) must be 2

eqs = [] if -2 in coeffs: eqs.append(sp.Eq(coeffs[-2], 0)) if -1 in coeffs: eqs.append(sp.Eq(coeffs[-1], 0)) eqs.append(sp.Eq(coeffs[0], 2))

print("Equations:", eqs) solutions = sp.solve(eqs, (a, b), dict=True) print("Solutions for a,b:", solutions)

If solution found, compute 100(a+b)^2

if solutions: sol = solutions[0] a_val = sol[a] b_val = sol[b] result = 100*(a_val + b_val)**2 print("a:", a_val, "b:", b_val, "100(a+b)^2:", sp.simplify(result)) else: print("No solutions found")

[THOUGHT] Using the series expansion and the conditions for the limit to be finite and equal to 2, the Python computation gives a = 24/5 and b = -12/5, leading to 100(a+b)^2 = 576. [ \boxed{576} ]