Problem 44 - Calculus Single Variable

Question

In each of Exercises \(41-54,\) determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{1} \ln ^{2}(x) d x\)

Step-by-Step Solution

Answer

The integral converges and evaluates to 2.

Step 1: Identify the Integral Type

The given integral \( \int_{0}^{1} \ln^{2}(x) \, dx \) is considered improper because the natural logarithm function \( \ln(x) \) approaches negative infinity as \( x \) approaches zero. We need to determine if this integral is convergent or divergent.

Step 2: Set Up the Limit to Analyze the Behavior at the Lower Bound

Since \( \ln(x) \) becomes problematic at \( x = 0 \), we set up a limit to address this issue:\[\lim_{a \to 0^+} \int_{a}^{1} \ln^{2}(x) \, dx.\]

Step 3: Integrate by Parts

To evaluate \( \int \ln^{2}(x) \, dx \), we use integration by parts, where \( u = \ln^{2}(x) \) and \( dv = dx \). Then, \( du = \frac{2\ln(x)}{x} \, dx \) and \( v = x \). Applying the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get:\[x \ln^{2}(x) \Big|_{a}^{1} - \int_{a}^{1} x \cdot \frac{2\ln(x)}{x} \, dx = x \ln^{2}(x) \Big|_{a}^{1} - 2\int_{a}^{1} \ln(x) \, dx.\]

Step 4: Evaluate Initial Setup of Integration by Parts

When \( x = 1 \), \( \ln^{2}(1) = 0 \), so the first term \( x \ln^{2}(x) \Big|_{a}^{1} = 0 - a \ln^{2}(a) \). Now, evaluate the remaining integral \( -2 \int_{a}^{1} \ln(x) \, dx \).

Step 5: Integrate the Remaining Integral

Integrate \( \int \ln(x) \, dx \) again by parts. Set \( u = \ln(x) \) and \( dv = dx \), then \( du = \frac{1}{x} \, dx \) and \( v = x \). This gives us:\[x\ln(x) \Big|_{a}^{1} - \int_{a}^{1} 1 \, dx = x\ln(x) \Big|_{a}^{1} - (x \big|_{a}^{1}).\]Thus, \(-2\left((1 \ln(1) - a \ln(a)) - (1 - a)\right)\).

Step 6: Substitute Back the Limits and Simplify

At \( x = 1 \), both \( 1 \ln(1) \) and \( 1 \) simplify to zero. At \( x = a \), terms involving \( a \) remain. Therefore, the integral becomes:\[-2[-a \ln(a) + a - 1].\] When evaluated as \( a \to 0^+ \), both \(-a \ln(a)\) and \(a\) approach zero, making the expression approach \(-2(-1) = 2\).

Step 7: Conclusion on Convergence and Evaluation

Since the final limit exists and equals \( 2 \), the integral converges and its value is \( 2 \).

Key Concepts

Understanding Convergence of Improper Integrals

Improper integrals are somewhat special in the family of integrals. They typically involve infinite limits or points where the integrand becomes unbounded. In the case of our integral, \( \int_{0}^{1} \ln^{2}(x) \, dx \), we're dealing with the latter. Here, the issue arises as the natural logarithm function dives towards negative infinity as \( x \) approaches zero from the right side.
To define whether an improper integral converges, we often replace the problematic point with a variable approaching the point. In this exercise, we look at:

• \( \lim_{a \to 0^+} \int_{a}^{1} \ln^{2}(x) \, dx \)

This limit helps dissect the behavior of the integral as \( x \) nears zero. If the limit proves finite, the improper integral is identified as convergent; otherwise, it is deemed divergent.
Through such limits, we effectively deal with infinities, making it easier to determine the behavior of improper integrals.

Integration by Parts: A Useful Technique

Integration by parts is a powerful tool when tackling integrals that don't immediately lend themselves to straightforward calculus techniques. The formula for integration by parts is:
\[ \int u \, dv = uv - \int v \, du \]
This approach works by breaking down the given integral into simpler parts. For our problem \( \int \ln^{2}(x) \, dx \), we strategically choose:

• \( u = \ln^{2}(x) \) (the term we wish to differentiate)
• \( dv = dx \) (the term we would integrate)

From this, we get \( du = \frac{2\ln(x)}{x} \, dx \) and \( v = x \).
Subsequently, applying the integration by parts formula, not just once, but successively, allows us to rewrite a more complex integral into parts we can more easily integrate.
This careful breakdown into manageable terms makes integration by parts an essential technique, allowing us to simplify and solve seemingly difficult integrals.

The Role of the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function with unique behaviors and properties. It's primarily defined for positive real numbers and is the inverse of the exponential function.
In calculus, the natural logarithm frequently appears in integration and differentiation, due to its practical mathematical properties. One crucial property is its approximation near zero, illustrating how the function moves towards negative infinity as \( x \) approaches zero from the positive side.
This tendency is a key aspect in analyzing and solving integrals like \( \int_{0}^{1} \ln^{2}(x) \, dx \). Here, understanding the behavior of \( \ln(x) \) provides insights into how to handle points where the function tends to infinity.
Thus, recognizing the nature and characteristics of the natural logarithm is vital in integrating complex functions and determining the convergence of integrals.