Question

Evaluate the integrals. If the integral diverges, answer "diverges."\(\int_{0}^{1} \frac{d x}{x^{\pi}}\)

Short answer

The integral diverges.

Step-by-step solution

Understanding the Problem

We need to determine if the integral \( \int_{0}^{1} \frac{dx}{x^{\pi}} \) converges or diverges. If it converges, we should find its value.

Analyze the Integrand

The integrand is \( \frac{1}{x^{\pi}} \), which becomes problematic at \( x = 0 \) because \( x^{\pi} \) approaches 0, making the integrand approach infinity.

Determine Convergence or Divergence

For \( x^{\pi} \, \text{with} \, \pi > 1 \), the integral \( \int_{a}^{b} x^{-n}\, dx \) converges for \( n < 1 \). Here we have \( n = \pi \), where \( \pi > 1 \). Thus, the integral \( \int_{0}^{1} x^{-\pi} \, dx \) diverges.

Conclusion

Since \( \pi > 1 \), the integral \( \int_{0}^{1} \frac{dx}{x^{\pi}} \) diverges. Hence, there is no finite value for this integral.

Key concepts

Convergence and Divergence

The concepts of convergence and divergence are fundamental in mathematical analysis. Evaluating whether an integral converges or diverges helps us understand if the integral tends to a finite value or not. In the case of improper integrals, the convergence or divergence depends significantly on the behavior of the integrand at the endpoints of integration.
To determine convergence or divergence, we often analyze the value of the power in the integrand. This is particularly important in integrals involving terms like \(x^{-n}\). If the power \(n\) is greater than 1, the integral from 0 to 1 tends to become unbounded as \(x\) approaches zero. Therefore, the integral diverges, which means it does not converge to any finite number. This is precisely what happened in the integral \( \int_{0}^{1} \frac{dx}{x^{\pi}} \) with \( \pi \) being approximately 3.14, which is much greater than 1.
In sums or integrals where convergence or divergence is not immediately apparent, applying comparison tests or the limit comparison test often helps assess the integral's behavior.

Power Integrals

Power integrals are integrals in which the integrand is a power function, usually of the form \(x^{-n}\). These types of integrals are essential when analyzing functions that involve exponential relationships. Understanding the power \(n\) is crucial. If \(n < 1\), the integral of \(x^{-n}\) over the interval from 0 to 1 converges, meaning it evaluates to a finite number. However, if \(n \geq 1\), the integral diverges. This divergence happens because the function's values grow too large as we approach zero, resulting in an infinite area under the curve. For example, consider the integral \( \int_{0}^{1} x^{-n} \, dx \):

• \(n = 0.5\), converges to a finite value.
• \(n = 1\), the integral diverges as it transforms to an integral like \(\ln(x)\).
• \(n = 2\), diverges because the power is greater than 1.

This behavior indicates how critical the power of \(x\) is in determining the outcome of an integral.

Mathematical Analysis

Mathematical analysis involves studying these integrals to understand their properties and behaviors. It goes beyond computation to consider the implications of convergence and divergence in both theoretical and practical contexts.
In the context of improper integrals, mathematical analysis requires a broad knowledge of functions and limits to ascertain if an integral has a finite solution. Utilizing integration techniques and understanding the type of function is pivotal. Techniques such as substitution, partial fractions, and limit comparisons all play a role in analyzing and simplifying these integrals.
Moreover, mathematical analysis emphasizes logic and proof, ensuring that conclusions about integrals are supported by solid reasoning. By examining the structure of integrals and their limits, we delve deeper into their behavior and characteristics, reinforcing our understanding of calculus and higher-level mathematics.