Question

Determine whether the following integrals converge or diverge. $$\int_{2}^{\infty} \frac{x^{3}}{x^{4}-x-1} d x$$

Short answer

Answer: The integral diverges.

Step-by-step solution

Define the given integral

We are given the integral: $$\int_{2}^{\infty} \frac{x^{3}}{x^{4}-x-1} d x$$

Analyze the function and find a suitable function to compare

As x goes to infinity, the dominant term in the denominator is \(x^4\). Therefore, we can simplify the given function to: $$\frac{x^{3}}{x^{4}-x-1} \approx \frac{x^{3}}{x^4}$$ Now, let's consider the function \(\frac{1}{x}\), so that we can compare our simplified function with: $$\frac{x^{3}}{x^{4}} = \frac{1}{x}$$

Apply the comparison test

We know that the integral of \(\frac{1}{x}\), that is, $$\int_{2}^{\infty} \frac{1}{x} d x$$ diverges, because it's an example of the p-series test for the convergence of improper integrals. Since \(\frac{1}{x}\) is the simplified version of our original function, we can apply the comparison test. As \(x\) goes to infinity, \(\frac{x^{3}}{x^{4}-x-1}\) and \(\frac{1}{x}\) share the same dominant term. It means they have the same behavior for large values of x, and therefore, the given integral has the same convergence properties as the integral of the simplified function. Since the integral of the simplified function \(\frac{1}{x}\) diverges, the given integral also diverges.

Conclusion

The integral $$\int_{2}^{\infty} \frac{x^{3}}{x^{4}-x-1} d x$$ diverges, based on the comparison test with the simplified function \(\frac{1}{x}\).

Key concepts

Comparison Test

The Comparison Test is a handy tool to determine the convergence or divergence of improper integrals. By using this test, we can compare a complex integral with a simpler, often well-known integral.
Improper integrals often stretch from a boundary to infinity, making their evaluation tricky. The key is to find another function that behaves similarly as the variable approaches infinity.
In our example, we simplified \(\frac{x^{3}}{x^{4}-x-1}\) to \(\frac{1}{x}\), recognizing that they share the same dominant term behavior for large \(x\). If the simpler function's integral diverges or converges, then the original integral will follow suit, given their similarity in terms.

Convergence and Divergence

One of the essential concepts in calculus is understanding whether an integral converges or diverges. This concept tells us if the area under a curve (from some point to infinity) results in a finite value or not.
For the integral of \(\frac{1}{x}\) from 2 to infinity, it diverges. This means the area under the curve continues to increase indefinitely.
Convergence means the integral has a finite value, while divergence means it does not. When using the Comparison Test, observing the behavior of the simpler function, like \(\frac{1}{x}\), gives us insight into the integral of the original, more complex function.

Dominant Term Analysis

Dominant Term Analysis is a strategy to simplify complex expressions, focusing on how terms impact the behavior of the function as the variable grows.
In our problem, \(\frac{x^{3}}{x^{4}-x-1}\), the term \(x^4\) is dominant in the denominator when \(x\) approaches infinity.

• The terms \(-x-1\) become insignificant for large values of \(x\).
• This allows us to approximate the function by ignoring less significant terms.

By simplifying to \(\frac{1}{x}\), we capture the "leading behavior" of the integral, making it possible to utilize tests like the Comparison Test effectively. This simplification is crucial for analyzing convergence or divergence efficiently.