To find the exact value of the integral [tex]\(\int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx\)[/tex], we can follow these steps:
### Step 1: Simplify the Integrand
First, simplify the integrand [tex]\(\frac{4}{\sqrt[3]{x^2}}\)[/tex]:
[tex]\[ \frac{4}{\sqrt[3]{x^2}} = 4 \cdot x^{-2/3} \][/tex]
### Step 2: Integrate the Function
Next, integrate [tex]\(4x^{-2/3}\)[/tex] using the power rule for integration:
[tex]\[
\int 4x^{-2/3} \, dx = 4 \int x^{-2/3} \, dx = 4 \left( \frac{x^{1 - 2/3}}{1 - 2/3} \right) = 4 \left( \frac{x^{1/3}}{1/3} \right) = 4 \cdot 3 x^{1/3} = 12 x^{1/3}
\][/tex]
### Step 3: Apply the Fundamental Theorem of Calculus
Evaluate this antiderivative at the bounds [tex]\(-2\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[
\int_{-2}^6 4 x^{-2/3} \, dx = \left. 12 x^{1/3} \right|_{-2}^6 = 12 \left( 6^{1/3} - (-2)^{1/3} \right)
\][/tex]
### Step 4: Simplify the Evaluated Result
Simplify the expression:
[tex]\[
12 \left( 6^{1/3} - (-2)^{1/3} \right)
\][/tex]
Here, [tex]\((-2)^{1/3}\)[/tex] can be expressed as [tex]\(- (2)^{1/3}\)[/tex], so:
[tex]\[
12 \left( 6^{1/3} - (-2)^{1/3} \right) = 12 (6^{1/3} + 2^{1/3})
\][/tex]
### Step 5: Match with Given Options
The exact value of the integral matches one of the given options:
[tex]\[
\boxed{12 (\sqrt[3]{6} + \sqrt[3]{2})}
\][/tex]
Thus, the correct answer is:
[tex]\[ 12 (\sqrt[3]{6} + \sqrt[3]{2}) \][/tex]
