Express $\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2}$ in partial fractions. Hence, find $\int \frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} \, dx$.

1. Perform long division to extract the polynomial part since the degree of the numerator equals the degree of the denominator.

$\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} = \frac{3(3x^2 + 5x - 2) - 15x + 6 + 17x - 16}{3x^2 + 5x - 2}$
$= 3 + \frac{2x - 10}{3x^2 + 5x - 2}$

2. Factorise the denominator of the proper fraction.

$3x^2 + 5x - 2 = (3x - 1)(x + 2)$

3. Set up the partial fraction decomposition for the proper fraction.

Let $\frac{2x - 10}{(3x - 1)(x + 2)} = \frac{A}{3x - 1} + \frac{B}{x + 2}$
$2x - 10 = A(x + 2) + B(3x - 1)$

4. Solve for the constants $A$ and $B$.

Substitute $x = -2$:
$2(-2) - 10 = B(3(-2) - 1)$
$-14 = -7B \implies B = 2$

Substitute $x = \frac{1}{3}$:
$2\left(\frac{1}{3}\right) - 10 = A\left(\frac{1}{3} + 2\right)$
$-\frac{28}{3} = \frac{7}{3}A \implies A = -4$

5. Write the final expression in partial fractions.

$\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} = 3 - \frac{4}{3x - 1} + \frac{2}{x + 2}$

1. Use the partial fraction expansion from part (a)

From part (a), we have:
$\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} = 3 - \frac{4}{3x - 1} + \frac{2}{x + 2}$

2. Integrate each term with respect to $x$

$\int \left( 3 - \frac{4}{3x - 1} + \frac{2}{x + 2} \right) \, dx$
$= 3x - \frac{4}{3}\ln(3x - 1) + 2\ln(x + 2) + C$