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Explain how and why you got the numbers and where they come from for all of the steps.

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

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### 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:
\[
\int (4x^2 + 4x + 9) \, dx = \frac{4}{3} x^3 + 2x^2 + 9x + C
\]

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### 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}

This page provides detailed, step-by-step solutions to a range of calculus problems involving anti-derivatives. Topics include applying the power rule, logarithmic integrals, and exponential integrals, with explanations for each calculation. Suitable for college-level calculus students.

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