[THOUGHT] I will expand the integral and x cos x near 0 as power series, enforce that the x-term cancels so the expression divided by x^3 has a finite limit, then match the x^3 coefficient to make the limit equal 2. I’ll verify the resulting a and b by computing the limit in Python exactly. [/THOUGHT] [PYTHON] import sympy as sp

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

expr = (1/x**3)((a/2)sp.integrate(1/(1-sp.Symbol('t')**2),(sp.Symbol('t'),0,x)) + bxsp.cos(x))

Better: use known integral: atanh(x)

expr2 = (1/x**3)((a/2)sp.atanh(x) + bxsp.cos(x)) series = sp.series((a/2)sp.atanh(x) + bx*sp.cos(x), x, 0, 5) print("Series numerator:", series)

Solve for cancellation and limit

coefficient of x is 0, coefficient of x^3 is 2

num_series = sp.series((a/2)sp.atanh(x) + bx*sp.cos(x), x, 0, 5).removeO() coeff1 = sp.expand(num_series).coeff(x,1) coeff3 = sp.expand(num_series).coeff(x,3) sol=sp.solve([sp.Eq(coeff1,0), sp.Eq(coeff3,2)], [a,b], dict=True) print("Solutions:", sol)

sol=sol[0] a_val=sp.nsimplify(sol[a]) b_val=sp.nsimplify(sol[b]) print("a,b:", a_val, b_val)

limit_val = sp.limit(expr2.subs({a:a_val,b:b_val}), x, 0) print("Limit check:", limit_val)

result = sp.simplify(100*(a_val+b_val)**2) print("100(a+b)^2 =", result) [/PYTHON]

[THOUGHT] The previous code errored; I’ll redo the computation with a correct atanh representation for the integral, solve for a and b from the series conditions, then compute 100(a+b)^2 and verify the limit in Python. [/THOUGHT] [PYTHON] import sympy as sp

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

integral from 0 to x of 1/(1-t^2) dt = atanh(x)

expr = (1/x**3)((a/2)sp.atanh(x) + bxsp.cos(x))

Series of numerator up to x^5

num = (a/2)sp.atanh(x) + bx*sp.cos(x) num_series = sp.series(num, x, 0, 6).removeO() coeff1 = sp.expand(num_series).coeff(x,1) coeff3 = sp.expand(num_series).coeff(x,3)

sol = sp.solve([sp.Eq(coeff1,0), sp.Eq(coeff3,2)], [a,b], dict=True)[0] a_val, b_val = sp.simplify(sol[a]), sp.simplify(sol[b])

print("a =", a_val, "b =", b_val) print("limit =", sp.simplify(sp.limit(expr.subs(sol), x, 0))) print("100(a+b)^2 =", sp.simplify(100*(a_val+b_val)**2)) [/PYTHON]

[THOUGHT] Using the verified values (a=\frac{24}{5}) and (b=-\frac{12}{5}), we get (100(a+b)^2=576). \boxed{576}