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To solve the integral

\[
\int_{-2}^{4} \left( \frac{x^4}{3} - \frac{1}{3x} + 2 \right) dx,
\]

we'll integrate each term in the integrand separately and then evaluate the definite integral over the interval \([-2, 4]\).

### Step 1: Break Down the Integral
Rewrite the integral as:
\[
\int_{-2}^{4} \frac{x^4}{3} \, dx - \int_{-2}^{4} \frac{1}{3x} \, dx + \int_{-2}^{4} 2 \, dx.
\]

### Step 2: Integrate Each Term Separately

#### 1. Integral of \(\frac{x^4}{3}\)
\[
\int \frac{x^4}{3} \, dx = \frac{1}{3} \int x^4 \, dx = \frac{1}{3} \cdot \frac{x^5}{5} = \frac{x^5}{15}.
\]

#### 2. Integral of \(-\frac{1}{3x}\)
\[
\int -\frac{1}{3x} \, dx = -\frac{1}{3} \int \frac{1}{x} \, dx = -\frac{1}{3} \ln |x|.
\]

#### 3. Integral of \(2\)
\[
\int 2 \, dx = 2x.
\]

### Step 3: Combine the Results
Now we have:
\[
\int_{-2}^{4} \left( \frac{x^4}{3} - \frac{1}{3x} + 2 \right) dx = \left[ \frac{x^5}{15} - \frac{1}{3} \ln |x| + 2x \right]_{-2}^{4}.
\]

### Step 4: Evaluate at the Limits \(x = 4\) and \(x = -2\)

#### At \(x = 4\):
\[
\frac{4^5}{15} - \frac{1}{3} \ln |4| + 2 \cdot 4 = \frac{1024}{15} - \frac{1}{3} \ln 4 + 8.
\]

#### At \(x = -2\):
Since \(\ln |x|\) includes the absolute value, it’s evaluated as \(x = 2\).
\[
\frac{(-2)^5}{15} - \frac{1}{3} \ln |2| + 2 \cdot (-2) = \frac{-32}{15} - \frac{1}{3} \ln 2 - 4.
\]

### Step 5: Compute the Final Value
Combine the terms from evaluating the limits:

\[
\int_{-2}^{4} \left( \frac{x^4}{3} - \frac{1}{3x} + 2 \right) dx = \left( \frac{1024}{15} - \frac{1}{3} \ln 4 + 8 \right) - \left( \frac{-32}{15} - \frac{1}{3} \ln 2 - 4 \right).
\]

Now, substitute \(\ln 4 = 2 \ln 2\), simplify, and perform the final arithmetic for an exact value.

Would you like further details on the final arithmetic steps, or have any questions?

---

1. How do you simplify logarithmic expressions in definite integrals?
2. Can definite integrals result in negative values?
3. Why is the absolute value used in \(\ln |x|\)?
4. What are potential challenges in evaluating piecewise integrals?
5. How does the Fundamental Theorem of Calculus apply here?

**Tip:** Always separate and handle each term in complex integrands to avoid calculation errors.

Evaluate the integral from -2 to 4 of (x^4 / 3 - 1 / (3x) + 2) dx.

This page provides a step-by-step solution for evaluating the definite integral from -2 to 4 of the function (x^4 / 3 - 1 / (3x) + 2). It covers concepts in calculus, including definite integrals, logarithmic functions, and the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12.

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