Let's work through each problem and find the anti-derivatives, step-by-step. I'll explain the reasoning and calculations behind each step.

Problem 1:

$\int (4x^2 + 4x + 9) \, dx$ For each term in the integrand:

1. $\int 4x^2 \, dx = \frac{4}{3} x^3$ since the power rule states $\int x^n \, dx = \frac{x^{n+1}}{n+1}$.
2. $\int 4x \, dx = 2x^2$, applying the power rule.
3. $\int 9 \, dx = 9x$, since the integral of a constant is the constant times $x$.

Putting it together: $\int (4x^2 + 4x + 9) \, dx = \frac{4}{3} x^3 + 2x^2 + 9x + C$

Problem 2:

$\int \left( \frac{5}{w} - 7e^w + 6\sqrt[3]{w} \right) \, dw$

1. $\int \frac{5}{w} \, dw = 5 \ln |w|$, since $\int \frac{1}{w} \, dw = \ln |w|$.
2. $\int -7e^w \, dw = -7e^w$, as the integral of $e^w$ is $e^w$.
3. ( \int 6w^{1/3} , dw = \frac{6}{4/3} w^{4/3} = \frac{18}{2} w^{4/3}