a) g(x) = 2x^2 + 5x + 6, ext{ where } x ext{ } ext{ in } ext{ R.} Find \int g(x) \, dx - Leaving Cert Mathematics - Question 2 - 2022

We start by integrating the function f(x) = ax^2 + bx + c:

$\int (ax^2 + bx + c) \, dx = \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C$
Evaluate this from the lower limit x_1 to the upper limit x_2 to find the area K:

$\left(\frac{a}{3}x^3 + \frac{b}{2}x^2 + cx\right) \bigg|_{x_1}^{x_2} = 538$
Substituting the limits in gives us:

$\left(\frac{a}{3}(x_2^3) + \frac{b}{2}(x_2^2) + cx_2\right) - \left(\frac{a}{3}(x_1^3) + \frac{b}{2}(x_1^2) + cx_1\right) = 538$
This expands to show the relationships for the coefficients a, b, and c.
We set up the resulting equation for K, and through previous integration and simplifications, we find:

$4a + 3b + 3c = 807$