Find the integral of 2x (x + 1/3) + 2x^2 - 3x + 7

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.

Learn how to integrate the polynomial function 2x(x + 1/3) + 2x^2 - 3x + 7 with a detailed step-by-step solution. This problem involves integration techniques for polynomials and covers key concepts such as the fundamental theorem of calculus and the constant of integration. Suitable for advanced high school students and college-level learners.

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