The minimum number of equal length subintervals needed to approximate ∫₁² x eˣ…

For the composite trapezoidal rule, the error bound is
|E_T| ≤ ((b - a)³ / (12n²)) max |f''(x)|.

Here f(x) = x eˣ on [1, 2].

f'(x) = (x + 1)eˣ
f''(x) = (x + 2)eˣ.

On [1, 2], f''(x) is increasing, so the maximum occurs at x = 2:
max |f''(x)| = 4e².

Since b - a = 1,
|E_T| ≤ (1/(12n²)) × 4e² = e²/(3n²).

We need this to be at most (1/3) × 10⁻⁶:
e²/(3n²) ≤ (1/3) × 10⁻⁶.

Multiplying by 3 gives
e²/n² ≤ 10⁻⁶,
so n² ≥ e² × 10⁶ and n ≥ 1000e.

Therefore, the required minimum number of equal subintervals is 1000e.