Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

Using the Fundamental Theorem of Calculus, find the exact value of [tex]\int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx[/tex].

A. [tex]12(\sqrt[3]{36}+\sqrt[3]{4})[/tex]

B. [tex]12(\sqrt[3]{36}-\sqrt[3]{4})[/tex]

C. [tex]6(\sqrt[3]{6}-\sqrt[3]{2})[/tex]

D. [tex]6(\sqrt[3]{6}+\sqrt[3]{2})[/tex]

E. [tex]12(\sqrt[3]{6}-\sqrt[3]{2})[/tex]

F. [tex]12(\sqrt[3]{6}+\sqrt[3]{2})[/tex]

Sagot :

To find the exact value of the definite integral [tex]\(\int_{-2}^6 \frac{4}{\sqrt[3]{x^2}} \, dx\)[/tex], we need to follow these steps using the Fundamental Theorem of Calculus.

1. Rewrite the integrand:

[tex]\[\frac{4}{\sqrt[3]{x^2}} = 4 \cdot x^{-2/3}\][/tex]

2. Find the antiderivative:

To find the antiderivative of [tex]\(4x^{-2/3}\)[/tex], we apply the integration rule for power functions:

[tex]\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1\][/tex]

Here, [tex]\(n = -\frac{2}{3}\)[/tex], so we get:

[tex]\[ \int 4x^{-2/3} \, dx = 4 \cdot \int x^{-2/3} \, dx = 4 \cdot \left(\frac{x^{(-2/3) + 1}}{(-2/3) + 1}\right) + C = 4 \cdot \left(\frac{x^{1/3}}{1/3}\right) + C = 4 \cdot 3x^{1/3} + C = 12x^{1/3} + C \][/tex]

3. Apply the limits:

Now we evaluate this antiderivative from [tex]\(-2\)[/tex] to [tex]\(6\)[/tex]:

[tex]\[ \left[ 12x^{1/3} \right]_{-2}^{6} = 12 \left(6^{1/3}\right) - 12 \left((-2)^{1/3}\right) \][/tex]

4. Analyze the cube roots:

The cube root of [tex]\(6\)[/tex] is straightforward. However, the cube root of [tex]\(-2\)[/tex] can be tricky. For real numbers, [tex]\((-2)^{1/3}\)[/tex] is defined as the real cube root of [tex]\(-2\)[/tex], which is approximately [tex]\(-1.2599\)[/tex].

Thus, we need to carefully consider the real part in this specific case.

Now, let's contrast the evaluated values:

[tex]\[ 12 \left(6^{1/3}\right) = 12 (1.817) \approx 21.804 \][/tex]

[tex]\[ 12 \left((-2)^{1/3}\right) = 12 (-1.259) = -15.108 \][/tex]

5. Summarize the integral value:

Summing these values:

[tex]\[ \left[ 12x^{1/3} \right]_{-2}^{6} = 12 \cdot 6^{1/3} - 12 \cdot (-2^{1/3}) = 12 \left(6^{1/3}\right) + 12 \left(2^{1/3}\right) = 12 \left( 6^{1/3} + 2^{1/3} \right) \][/tex]

From the given options, the correct answer is:
[tex]\[ \boxed{12(\sqrt[3]{6}+\sqrt[3]{2})} \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.