Determine Whether The Integral Is Convergent Or Divergent. If It Is Convergent, Evaluate It. (If The Quantity Diverges, Enter DIVERGES.)$\int_{-2}^{14} \frac{8}{\sqrt[4]{x+2}} \, Dx$

Introduction

In this article, we will explore the convergence or divergence of the given integral, $\int_{-2}^{14} \frac{8}{\sqrt[4]{x+2}} \, dx$. To determine whether the integral is convergent or divergent, we need to analyze the behavior of the integrand as $x$ approaches the limits of integration.

Understanding the Integrals

The given integral is a definite integral, which means it has a specific upper and lower limit of integration. In this case, the lower limit is $-2$ and the upper limit is $14$. The integrand is a rational function of the form $\frac{8}{\sqrt[4]{x+2}}$.

Convergence or Divergence Criteria

To determine whether the integral is convergent or divergent, we need to apply the convergence or divergence criteria. There are several criteria that can be used to determine convergence or divergence, including:

• The Integral Test: This test states that if the function $f(x)$ is continuous and positive on the interval $[a, \infty)$, then the integral $\int_{a}^{\infty} f(x) \, dx$ converges if and only if the series $\sum_{n=a}^{\infty} f(n)$ converges.
• The Comparison Test: This test states that if $f(x) \leq g(x)$ for all $x$ in the interval $[a, b]$, and the integral $\int_{a}^{b} g(x) \, dx$ converges, then the integral $\int_{a}^{b} f(x) \, dx$ also converges.
• The Limit Comparison Test: This test states that if the limit $\lim_{x\to\infty} \frac{f(x)}{g(x)}$ exists and is finite, then the integral $\int_{a}^{\infty} f(x) \, dx$ converges if and only if the integral $\int_{a}^{\infty} g(x) \, dx$ converges.

Applying the Convergence or Divergence Criteria

In this case, we can apply the Limit Comparison Test to determine whether the integral is convergent or divergent. We can compare the integrand $\frac{8}{\sqrt[4]{x+2}}$ with the function $f(x) = \frac{1}{x^{\frac{1}{4}}}$.

Step 1: Find the Limit

To apply the Limit Comparison Test, we need to find the limit $\lim_{x\to\infty} \frac{\frac{8}{\sqrt[4]{x+2}}}{\frac{1}{x^{\frac{1}{4}}}}$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the integrand
integrand = 8 / (x + 2)**(1/4)

# Define the function
f = 1 / x**(1/4)

# Find the limit
limit = sp.limit(integrand / f, x, sp.oo)

print(limit)

The limit is equal to $8$.

Step 2: Determine Convergence or Divergence

Since the limit is finite and equal to $8$, we can conclude that the integral $\int_{-2}^{14} \frac{8}{\sqrt[4]{x+2}} \, dx$ converges if and only if the integral $\int_{-2}^{14} \frac{1}{x^{\frac{1}{4}}} \, dx$ converges.

Step 3: Evaluate the Integral

To evaluate the integral, we can use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the integrand
integrand = 1 / x**(1/4)

# Evaluate the integral
integral = sp.integrate(integrand, x)

print(integral)

The integral is equal to $4x^{\frac{3}{4}} + C$.

Conclusion

In conclusion, the integral $\int_{-2}^{14} \frac{8}{\sqrt[4]{x+2}} \, dx$ converges and is equal to $4(14)^{\frac{3}{4}} - 4(-2)^{\frac{3}{4}}$.

Final Answer

Introduction

In our previous article, we explored the convergence or divergence of the given integral, $\int_{-2}^{14} \frac{8}{\sqrt[4]{x+2}} \, dx$. We applied the Limit Comparison Test to determine whether the integral is convergent or divergent. In this article, we will answer some frequently asked questions related to determining convergence or divergence of the given integral.

Q: What is the Limit Comparison Test?

A: The Limit Comparison Test is a test used to determine whether an improper integral converges or diverges. It states that if the limit $\lim_{x\to\infty} \frac{f(x)}{g(x)}$ exists and is finite, then the integral $\int_{a}^{\infty} f(x) \, dx$ converges if and only if the integral $\int_{a}^{\infty} g(x) \, dx$ converges.

Q: How do I apply the Limit Comparison Test?

A: To apply the Limit Comparison Test, you need to:

1. Find the limit $\lim_{x\to\infty} \frac{f(x)}{g(x)}$.
2. Determine whether the limit is finite and equal to a non-zero value.
3. If the limit is finite and equal to a non-zero value, then the integral $\int_{a}^{\infty} f(x) \, dx$ converges if and only if the integral $\int_{a}^{\infty} g(x) \, dx$ converges.

Q: What is the power rule of integration?

A: The power rule of integration is a rule used to integrate functions of the form $x^n$. It states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$.

Q: How do I evaluate the integral using the power rule of integration?

A: To evaluate the integral using the power rule of integration, you need to:

1. Identify the function $f(x)$ that you want to integrate.
2. Determine the value of $n$ in the function $f(x) = x^n$.
3. Apply the power rule of integration to find the integral: $\int f(x) \, dx = \frac{x^{n+1}}{n+1} + C$.

Q: What is the final answer to the given integral?

A: The final answer to the given integral is $\boxed{4(14)^{\frac{3}{4}} - 4(-2)^{\frac{3}{4}}}$.

Q: Can I use the Limit Comparison Test to determine convergence or divergence of any integral?

A: No, the Limit Comparison Test can only be used to determine convergence or divergence of improper integrals. It cannot be used to determine convergence or divergence of definite integrals.

Q: What are some common mistakes to avoid when applying the Limit Comparison Test?

A: Some common mistakes to avoid when applying the Limit Comparison Test include:

• Not checking whether the limit is finite and equal to a non-zero value.
• Not determining whether the integral $\int_{a}^{\infty} g(x) \, dx$ converges or diverges.
• Not applying the power rule of integration correctly.

Conclusion

In conclusion, the Limit Comparison Test is a powerful tool used to determine convergence or divergence of improper integrals. By applying the Limit Comparison Test, we can determine whether an integral converges or diverges. We hope that this Q&A article has provided you with a better understanding of the Limit Comparison Test and how to apply it to determine convergence or divergence of the given integral.