Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{10}^{\infty} \frac{x}{1+x^{2}} d x\)

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Identify the Type of Improper Integral

The given integral is \(\int_{10}^{\infty} \frac{x}{1+x^{2}} \, dx\). Notice the upper limit of integration is infinity, making it an improper integral of the first type.

Step 2: Set up the Limit for the Integral

To evaluate the integral, we replace infinity with a variable, say \(b\), and evaluate the limit. The integral becomes: \[ \lim_{b \to \infty} \int_{10}^{b} \frac{x}{1+x^{2}} \, dx \]

Step 3: Solve the Integral

Evaluate the integral \(\int \frac{x}{1+x^{2}} \, dx\) using substitution. Let \(u = 1 + x^2\), then \(du = 2x \, dx\), or \(\frac{1}{2}du = x \, dx\). The integral becomes: \[ \frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \, \ln|u| + C \] Substituting back for \(u\), we get: \[ \frac{1}{2} \, \ln|1 + x^2| + C \]

Step 4: Evaluate the Indefinite Integral from 10 to b

Now, plug in the limits 10 and \(b\) into the indefinite integral: \[ \frac{1}{2} \ln|1+b^2| - \frac{1}{2} \ln|1+10^2| \] This simplifies to: \[ \frac{1}{2} \ln|1+b^2| - \frac{1}{2} \ln|101| \]

Step 5: Evaluate the Limit as b Approaches Infinity

Now, evaluate: \(\lim_{b \to \infty} \frac{1}{2} \ln|1+b^2| - \frac{1}{2} \ln|101|\). As \(b\) tends to infinity, \(\ln|1 + b^2|\) approaches infinity, thereby making the entire expression: \[ \lim_{b \to \infty} \left( \frac{1}{2} \ln|1+b^2| - \frac{1}{2} \ln|101| \right) = \infty \]

Step 6: Conclusion on Convergence

Since the limit evaluates to infinity, the improper integral: \(\int_{10}^{\infty} \frac{x}{1+x^{2}} \, dx\) diverges.

Key Concepts

Integration by Substitution

Integration by substitution is a commonly used technique in calculus to simplify the process of calculating integrals. This method works by changing variables to transform a given integral into a simpler form that is easier to evaluate. It is particularly useful when the integrand is a compound function.

In the given problem, we needed to evaluate the improper integral \( \int \frac{x}{1+x^{2}} \, dx \). By using substitution, we can simplify this integral significantly. We chose \( u = 1 + x^2 \). This selection is strategic because the derivative of \( x^2 \), which is \( 2x \), closely matches the numerator \( x \) in the integrand.

• The differential \( du \) is equal to \( 2x \, dx \), which implies that \( x \, dx \) can be replaced with \( \frac{1}{2} du \).
• Rewriting the integrand in terms of \( u \) gives us \( \frac{1}{2} \int \frac{1}{u} \, du \).
• This integral can now be easily evaluated to \( \frac{1}{2} \ln|u| + C \).

Substituting back for \( u \) in terms of \( x \) yields \( \frac{1}{2} \ln|1 + x^2| + C \). This new form is much simpler to deal with, especially when applying limits, as we will see in the next steps.

Limit Evaluation

Limit evaluation is a crucial part of solving improper integrals. Since we are dealing with an infinite upper limit, we must express our integral in terms of a limit.

In the given exercise, we replace infinity with a variable \( b \), and examine the behavior of the integral as \( b \) approaches infinity. This step converts the integral into a finite form that is:

• \( \lim_{b \to \infty} \int_{10}^{b} \frac{x}{1+x^2} \, dx \)

We previously found the indefinite integral, \( \frac{1}{2} \ln|1+x^2| + C \). Now, by applying the limits to this evaluated form from 10 to \( b \), we get:

• \( \frac{1}{2} \ln|1+b^2| - \frac{1}{2} \ln|101| \)

Understanding the behavior of this expression as \( b \to \infty \) is the goal.

It is important to recognize that as \( b \) grows, the expression \( \ln|1+b^2| \) tends to infinity, leading us to our next concept.

Divergence of Integrals

The concept of divergence in integrals deals with their tendency to approach infinity rather than settling at a finite value. Determining whether an integral converges or diverges is essential in understanding the behavior of functions over an unbounded domain.

For the provided integral \( \int_{10}^{\infty} \frac{x}{1+x^2} \, dx \), we applied limit evaluation to understand its behavior as the upper bound approaches infinity. The expression resulted in:

• \( \lim_{b \to \infty} \Big( \frac{1}{2} \ln|1+b^2| - \frac{1}{2} \ln|101| \Big) = \infty \)

This concludes that the integral does not converge to a finite number. Instead, it grows indefinitely larger as \( b \to \infty \).

In simple terms, divergence indicates that the area under the curve defined by the function \( \frac{x}{1+x^2} \) from 10 to infinity is infinite.

This outcome is important in determining the boundaries and limitations of a function's integral across an unbounded domain.