[THOUGHT] We need to evaluate the integral

where $P(x)$ is a polynomial of degree 7. The integrand is analytic except at $x = 3i$ (pole of order 8). Since the integration path is along the real axis, we can close the contour in the upper half-plane if the integrand decays sufficiently for large $|x|$ in the upper half-plane. The factor $e^{ix}$ decays in the upper half-plane because $e^{ix} = e^{i(x+iy)} = e^{ix - y}$ with $y>0$. So we can close the contour upward. Then by the residue theorem, $I = 2\pi i \cdot \text{Residue at } x = 3i$ (since the pole is inside the contour? Wait: $x = 3i$ is in the upper half-plane (positive imaginary part), yes, because $3i$ has imaginary part 3 > 0. So indeed the pole is inside the closed contour (upper half-plane). So

Since the denominator is $(x-3i)^8$, the residue for a pole of order $n$ is given by

Here $n=8$, so

Thus

We need to compute the 7th derivative of $P(x) e^{ix}$ evaluated at $x=3i$. Since $P(x)$ is a polynomial of degree 7, its 7th derivative will be constant (the leading term's derivative). Let's compute.

Let $f(x) = P(x) e^{ix}$. Use Leibniz rule:

But $(e^{ix})^{(m)} = i^m e^{ix}$. So

Evaluate at $x=3i$.

Alternatively, note that $P(x)$ is degree 7, so $P^{(7)}(x) = 7! \cdot$ leading coefficient. The leading term of $P(x)$ is $x^7$ (coefficient 1). So $P^{(7)}(x) = 7!$. Lower derivatives depend on polynomial coefficients.

We can compute explicitly using Python symbolic differentiation.

We'll compute $P(x)$ as given, then compute the 7th derivative of $P(x) e^{ix}$ symbolically and evaluate at $x=3i$. Then multiply by $2\pi i / 7!$ to get $I$. Then extract real and imaginary parts $a$ and $b$, compute floor(10a) and floor(b), and sum them.

Let's do it step by step in Python. [/THOUGHT] [PYTHON] import sympy as sp

Define symbols

x = sp.symbols('x') i = sp.I # imaginary unit

Define polynomial 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)

Define f(x) = P(x) * exp(i*x)

f = P * sp.exp(i*x)

Compute 7th derivative

f7 = sp.diff(f, x, 7)

Evaluate at x = 3i

val = sp.simplify(f7.subs(x, 3*i)) print("7th derivative at 3i:", val)

Compute residue: val / 7!

residue = val / sp.factorial(7) print("Residue:", residue)

Compute integral I = 2pii * residue

I = 2sp.pii * residue print("Integral I:", I) print("I simplified:", sp.simplify(I))

Extract real and imaginary parts

a = sp.re(I) b = sp.im(I) print("a =", a) print("b =", b)

Compute floor(10a) and floor(b)

floor_10a = sp.floor(10*a) floor_b = sp.floor(b) print("floor(10a) =", floor_10a) print("floor(b) =", floor_b)

Compute sum

result = floor_10a + floor_b print("Result:", result) [/PYTHON]

[THOUGHT] From the Python output, we have:

• $a = \frac{134\pi e^{-3}}{35}$
• $b = \frac{5479\pi e^{-3}}{420}$
• $\lfloor 10a \rfloor = 5$
• $\lfloor b \rfloor = 2$
• Result = $5 + 2 = 7$

Thus the required integer is 7. [/THOUGHT] $\boxed{7}$