[THOUGHT] We use Cauchy's theorem, noting that the polynomial $f(z) = z^2$ is entire, making the integral over the closed contour $\Gamma$ equal to zero. Thus, $\oint_\Gamma f(z)\,dz = 0$, implying $\int_C f(z)\,dz = -\int_L f(z)\,dz$. For $L$, parametrize as $f(x) = x^2$ with $x$ from $-5/2$ to $5/2$:

, with co-prime $p=125$, $q=12$, leading to $p+q = 137$.

[/THOUGHT]

[PYTHON]

Calculate the real integral

from sympy import Symbol, integrate

x = Symbol('x') integral_L = integrate(x**2, (x, -5/2, 5/2)) p = int(integral_L.numerator) q = int(integral_L.denominator) result = -(p + q)

(p, q, result)

[/PYTHON]

Final Answer $\boxed{137}$.[THOUGHT] We use Cauchy's theorem, noting that the polynomial $f(z) = z^2$ is entire, making the integral over the closed contour $\Gamma$ equal to zero. Thus, $\oint_\Gamma f(z)\,dz = 0$, implying $\int_C f(z)\,dz = -\int_L f(z)\,dz$. For $L$, parametrize as $f(x) = x^2$ with $x$ from $-5/2$ to $5/2$:

, with co-prime $p=125$, $q=12$, leading to $p+q = 137$.

[/THOUGHT]

[PYTHON]

Calculate the real integral

from sympy import Symbol, integrate

x = Symbol('x') integral_L = integrate(x**2, (x, -5/2, 5/2)) p = int(integral_L.numerator) q = int(integral_L.denominator) result = -(p + q)

(p, q, result)

[/PYTHON]

Final Answer $\boxed{137}$.

[THOUGHT] Analytic computation corrects Python error. The integral $\int_{-5/2}^{5/2} x^2 dx$ equals $\frac{125}{12}$, hence $-\frac{125}{12}$ implies $p=125$, $q=12$. Their sum is $137$. Prior AttributeErrors arose from improper result handling, overridden accurately by analysis. [/THOUGHT] $\boxed{137}$[THOUGHT] Analytic computation corrects Python error. The integral $\int_{-5/2}^{5/2} x^2 dx$ equals $\frac{125}{12}$, hence $-\frac{125}{12}$ implies $p=125$, $q=12$. Their sum is $137$. Prior AttributeErrors arose from improper result handling, overridden accurately by analysis. [/THOUGHT] $\boxed{137}$