Problem 32 - Biocalculus Calculus for the Life Sciences

Question

For what values of \(p\) is the integral $$\int_{1}^{\infty} \frac{1}{x^{p}} d x$$ convergent? Evaluate the integral for those values of \(p\)

Step-by-Step Solution

Answer

The integral converges for \( p > 1 \) and evaluates to \( \frac{1}{p-1} \).

Step 1: Identify the Type of Integral

The given integral \( \int_{1}^{\infty} \frac{1}{x^{p}} \, dx \) is an improper integral, as it has an infinite limit of integration. We need to determine when this integral converges.

Step 2: Set Up the Integral with Limits

Rewrite the integral with a limit to handle the infinity: \[ \int_{1}^{\infty} \frac{1}{x^{p}} \, dx = \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{p}} \, dx. \] Here, \( b \) will approach infinity as we evaluate the integral.

Step 3: Consider Different Cases for p

The convergence of the integral depends on the value of \( p \). We need to analyze different cases, in particular: \( p = 1 \), \( p < 1 \), and \( p > 1 \).

Step 4: Evaluate the Integral for p != 1

For \( p eq 1 \), the antiderivative of \( \frac{1}{x^p} \) is \( \frac{x^{1-p}}{1-p} \). Evaluate \[ \int_{1}^{b} \frac{1}{x^p} \, dx = \left[ \frac{x^{1-p}}{1-p} \right]_{1}^{b} = \frac{b^{1-p}}{1-p} - \frac{1^{1-p}}{1-p}. \]

Step 5: Evaluate the Limit for p > 1

For \( p > 1 \), \( 1-p < 0 \), so \( b^{1-p} \to 0 \) as \( b \to \infty \). Thus, \[ \lim_{b \to \infty} \left( \frac{b^{1-p}}{1-p} - \frac{1^{1-p}}{1-p} \right) = 0 - \frac{1}{1-p} = \frac{1}{p-1}. \] Therefore, the integral converges and evaluates to \( \frac{1}{p-1} \).

Step 6: Analyze Other Cases

For \( p = 1 \), the integral \( \int_{1}^{\infty} \frac{1}{x} \, dx \) diverges, as it is equivalent to \( \ln(b) \) which approaches infinity as \( b \to \infty \). For \( p < 1 \), the exponent \( 1-p > 0 \) implies that \( b^{1-p} \to \infty \) as \( b \to \infty \), hence the integral diverges.

Key Concepts

Convergence of Integrals

When dealing with integrals, convergence refers to whether the integral results in a finite value. For improper integrals, such as the one with infinite limits, determining convergence is crucial.
To assess the convergence of \( \int_{1}^{\infty} \frac{1}{x^{p}} \, dx \), we analyze the behavior of the function as it approaches infinity.
With different values of the parameter \( p \), the integral might either converge to a specific number or diverge, meaning it becomes unbounded.

• If \( p > 1 \), the integral converges and can be calculated, as the function behaves well towards the infinite end.
• If \( p = 1 \), the integral diverges due to the logarithmic nature causing it to grow indefinitely.
• If \( p < 1 \), again, the integral diverges. In this case, the evaluated function becomes more substantial as \( x \) grows, leading to infinite values.

Understanding convergence helps in evaluating whether a meaningful result exists for an integral.

Antiderivatives

An antiderivative of a function is a function whose derivative equals the original function. Finding antiderivatives is essential for computing integrals.
In the case of the integral \( \int_{1}^{b} \frac{1}{x^{p}} \, dx \), knowing its antiderivative allows us to evaluate the integral.
For \( p eq 1 \), the antiderivative of \( \frac{1}{x^{p}} \) is given by:

• \( \frac{x^{1-p}}{1-p} \)

This formula becomes vital when transitioning from definite integrals to evaluated results. You can see it applied in the results from \( x = 1 \) to some parameter \( b \). The term \( p = 1 \) is special because the integral becomes the natural logarithm, which requires separate consideration.
Antiderivatives thus provide the functional means to evaluate integrals consistently across various scenarios.

Limits of Integration

The limits of integration define where an integral starts and ends. Specifically, for our improper integral \( \int_{1}^{\infty} \frac{1}{x^{p}} \, dx \), the lower limit is 1 and the upper limit tends towards infinity.
The concept is different when compared to standard integrals with finite limits. In this scenario, employing a limit as a mathematical tool becomes necessary to replace the infinite boundary:

• Use \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{p}} \, dx \) to properly account for the infinite nature.

Setting the integral with a limit at the infinity end enables the determination of convergence effectively.
The change in limits from a finite number to infinity introduces the need for careful analysis, making the comprehension of this concept vital for correctly evaluating improper integrals.

Infinite Integrals

Infinite integrals, or improper integrals, feature at least one integration limit as infinity, such as our example \( \int_{1}^{\infty} \frac{1}{x^{p}} \, dx \).
These integrals often appear in calculus when evaluating areas under unbounded curves or for certain probability distribution functions. Working with infinite integrals involves approximating the behavior as one of the bounds tends to infinity.

• Convert the integral: \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^{p}} \, dx \).
• Analyze special scenarios, such as when \( p > 1 \), \( p = 1 \), or \( p < 1 \).

For a practical example, the convergence and evaluation depend on the parameter \( p \).
The ability to discern between converging or diverging cases helps provide valuable solutions in various mathematical and real-life applications, recognizing their importance in theory and practice.