To find the integral of the function $f(x) = 2x(x + \frac{1}{3}) + 2x^2 - 3x + 7$, let's first simplify the function.

First, distribute $2x$ in the term $2x(x + \frac{1}{3})$: $2x(x + \frac{1}{3}) = 2x^2 + \frac{2x}{3}$

Now, add this to the other terms in the function: $f(x) = 2x^2 + \frac{2x}{3} + 2x^2 - 3x + 7$

Combine like terms: $f(x) = 2x^2 + 2x^2 + \frac{2x}{3} - 3x + 7$ $f(x) = 4x^2 + \frac{2x}{3} - 3x + 7$

Now, let's integrate $f(x)$:

$\int (4x^2 + \frac{2x}{3} - 3x + 7) \, dx$

Integrate term by term:

1. $\int 4x^2 \, dx = \frac{4x^3}{3}$
2. $\int \frac{2x}{3} \, dx = \frac{2}{3} \cdot \frac{x^2}{2} = \frac{x^2}{3}$
3. $\int -3x \, dx = -\frac{3x^2}{2}$
4. $\int 7 \, dx = 7x$

Combine the integrals: $\int (4x^2 + \frac{2x}{3} - 3x + 7) \, dx = \frac{4x^3}{3} + \frac{x^2}{3} - \frac{3x^2}{2} + 7x + C$

Therefore, the integral of $2x(x + \frac{1}{3}) + 2x^2 - 3x + 7$ is: $\int (2x(x + \frac{1}{3}) + 2x^2 - 3x + 7) \, dx = \frac{4x^3}{3} + \frac{x^2}{3} - \frac{3x^2}{2} + 7x + C$

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How do you integrate a polynomial function?
2. What is the general formula for integrating $ax^n$?
3. How can you simplify algebraic expressions before integrating?
4. What are common integration techniques for different types of functions?
5. How do you find the constant of integration $C$?

Tip: Always combine like terms and simplify the function as much as possible before integrating to make the process easier.