Problem 66 - Thomas Calculus

Question

\(\int_{-\infty}^{\infty} f(x) d x\) may not equal \(\lim _{b \rightarrow \infty} \int_{-b}^{b} f(x) d x\) Show that $$\int_{0}^{\infty} \frac{2 x d x}{x^{2}+1}$$ diverges and hence that $$\int_{-\infty}^{\infty} \frac{2 x d x}{x^{2}+1}$$ diverges. Then show that $$\lim _{b \rightarrow \infty} \int_{-b}^{b} \frac{2 x d x}{x^{2}+1}=0$$

Step-by-Step Solution

Answer

The integral \(\int_{0}^{\infty} \frac{2x \, dx}{x^2+1}\) diverges, and thus \(\int_{-\infty}^{\infty} \frac{2x \, dx}{x^2+1}\) diverges. However, \(\lim_{b \rightarrow \infty} \int_{-b}^{b} \frac{2x \, dx}{x^2+1} = 0\).

Step 1: Analyze the Integral on (0, ∞)

We are given the integral \(\int_{0}^{\infty} \frac{2x \, dx}{x^{2}+1}\). Since the function \(\frac{2x}{x^2+1}\) behaves like \(\frac{2}{x}\) as \(x\) approaches infinity, the integral can be analyzed for divergence. Recognize that \(\int \frac{2}{x} \, dx\) diverges.

Step 2: Compute Improper Integral for (0, ∞)

Compute the antiderivative of \(\frac{2x}{x^2+1}\). The antiderivative is \(\ln(x^2+1)\). Evaluate the improper integral: \[ \int_{0}^{b} \frac{2x \, dx}{x^2+1} = \left[ \ln(x^2+1) \right]_{0}^{b} = \ln(b^2+1) - \ln(1) = \ln(b^2+1) \]As \(b\) approaches infinity, \(\ln(b^2+1)\) approaches infinity, indicating that the integral diverges.

Step 3: Conclusion About the Divergence of Original Integral

As the integral \(\int_{0}^{\infty} \frac{2x \, dx}{x^2+1}\) diverges, the integral over the whole real line \(\int_{-\infty}^{\infty} \frac{2x \, dx}{x^2+1}\) also diverges. This is because the unboundedness in the positive direction affects the convergence of the entire integral.

Step 4: Evaluate Limit of Symmetric Integral

Evaluate the integral \(\lim_{b \to \infty} \int_{-b}^{b} \frac{2x \, dx}{x^2+1}\). Using symmetry, observe that the function is odd: \(f(-x) = -f(x)\). Therefore:\[ \int_{-b}^{b} \frac{2x \, dx}{x^2+1} = 0 \]As \(b\) approaches infinity, the limit remains \(0\), as the contributions from \(-b\) to \(0\) and from \(0\) to \(b\) cancel each other.

Key Concepts

Divergence of Improper Integrals

Divergence is a concept that tells us whether an integral reaches infinity or not. In the exercise, you're dealing with an integral over an infinite domain, which leads to improper integrals. These are mathematical expressions that require limits to evaluate the area under a curve that stretches to infinity.

To understand divergence, consider the improper integral \[\int_{0}^{\infty} \frac{2x \, dx}{x^{2}+1}.\]Here, as \(x\) approaches infinity, the function \(\frac{2x}{x^2+1}\) roughly behaves like \(\frac{2}{x}\), a function whose integral is known to diverge. This means the area under the function between zero and infinity is endless.

• This type of function doesn't yield a finite value after integration.
• The integral diverges because the result heads towards infinity, rather than settling on a specific number.

Recognizing divergence is crucial because it helps determine whether integrals have finite values or not, impacting conclusions drawn from integrations over infinite intervals.

Understanding Antiderivatives

Antiderivatives are the "reverse" of derivatives. They allow us to find functions from known rates of change. Calculating an antiderivative is a key step in finding the value of a definite integral.

For the integral \[\int_{0}^{\infty} \frac{2x \, dx}{x^2+1},\]we identify that an antiderivative is \(\ln(x^2+1)\). By evaluating this from 0 to \(b\), where \(b\) moves towards infinity, you get:\[\ln(b^2+1) - \ln(1).\]As \(b\) goes to infinity, \(\ln(b^2+1)\) increases indefinitely, confirming the divergence.

• Antiderivatives help us translate complex functions into more manageable forms for solving integrals.
• They reveal how the accumulated quantity changes over an interval.

Grasping antiderivatives is essential for solving integrals, particularly in scenarios with infinite limits.

Symmetry in Functions and Its Impact

Symmetry often simplifies mathematical problems. In this context, the symmetry of the function \(f(x) = \frac{2x}{x^2+1}\) as an odd function plays a pivotal role.

An odd function fulfills \(f(-x) = -f(x)\). This property showcases a kind of symmetry about the origin, with equal but opposite heights for the function on either side of the y-axis.

When evaluating\[\int_{-b}^{b} \frac{2x \, dx}{x^2+1},\]the symmetry implies that the integral over the negative half of the domain will cancel out the integral over the positive half, yielding a total value of zero if \(b\) tends towards infinity.

• Symmetry can be utilized to predict or simplify outcomes of integrals.
• Knowing a function's symmetry helps in determining integral values more efficiently.

Understanding function symmetry enables effective analysis of integral properties, and is especially beneficial in evaluating limits and integral convergence.