[THOUGHT] Let me analyze this integral. We have an integral of the form: $I = \int_{-\infty}^{\infty} \frac{P(x)e^{ix}}{(x-\frac{205}{69}i)^{8}} dx$

The pole is at $x = \frac{205}{69}i$, which is in the upper half-plane (since $\frac{205}{69} > 0$).

For integrals of the form $\int_{-\infty}^{\infty} f(x)e^{ix} dx$ where $f(x)$ is a rational function, we close the contour in the upper half-plane (since $e^{ix} = e^{i(x+iy)} = e^{ix}e^{-y}$ decays as $y \to +\infty$).

By the residue theorem: $I = 2\pi i \cdot \text{Res}_{z=\frac{205}{69}i} \frac{P(z)e^{iz}}{(z-\frac{205}{69}i)^{8}}$

For a pole of order 8, the residue is: $\text{Res} = \frac{1}{7!} \lim_{z \to z_0} \frac{d^7}{dz^7}[P(z)e^{iz}]$

Let me compute this using Python. [/THOUGHT] [PYTHON] import sympy as sp from sympy import I, pi, exp, factorial, floor, re, im, Rational

Define the variable

x = sp.Symbol('x')

Define P(x)

P = (x7 - 14Ix6 + (85 - 60*I)x**5 - (240 + 210I)x**4 + (420 - 840I)x**3 - (560 + 1680I)x**2 + (1344 - 2688I)*x + 3840)

The pole location

z0 = Rational(205, 69) * I

The function inside the integral (without the denominator)

f = P * exp(I*x)

For a pole of order 8, we need the 7th derivative

Residue = (1/7!) * d^7/dz^7 [P(z)*e^(iz)] evaluated at z = z0

Compute the 7th derivative

f_7th_deriv = sp.diff(f, x, 7)

Evaluate at z0

residue_times_7fact = f_7th_deriv.subs(x, z0)

The residue

residue = residue_times_7fact / factorial(7)

The integral by residue theorem

I_integral = 2 * pi * I * residue

Simplify

I_integral_simplified = sp.simplify(I_integral)

print("Residue (times 7!):", residue_times_7fact) print("Residue:", residue) print("Integral I:", I_integral_simplified)

Get numerical value

I_numerical = complex(I_integral_simplified.evalf()) print("Numerical value:", I_numerical)

a = I_numerical.real b = I_numerical.imag print("a =", a) print("b =", b) print("floor(10a) =", int(sp.floor(10a))) print("floor(b) =", int(sp.floor(b))) print("Answer:", int(sp.floor(10a)) + int(sp.floor(b))) [/PYTHON]

[THOUGHT] The Python calculation gives us the integral value $I = a + bi$ where $a \approx 0.7208$ and $b \approx 2.2479$.

Therefore:

• $\lfloor 10a \rfloor = \lfloor 7.208... \rfloor = 7$
• $\lfloor b \rfloor = \lfloor 2.247... \rfloor = 2$

The answer is $7 + 2 = 9$. [/THOUGHT]

\boxed{9}