Problem 43 - Calculus Single Variable

Question

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{\infty} \frac{1}{x^{4}+3 x^{2}+2} d x $$

Step-by-Step Solution

Answer

The integral converges and its value is \( \frac{\pi}{2} \).

Step 1: Analyze the Integrand

The integrand is \( \frac{1}{x^4 + 3x^2 + 2} \). Notice that \( x^4 + 3x^2 + 2 \) is a polynomial and its roots can help us understand its behavior. This can be factored as \( (x^2 + 1)(x^2 + 2) \).

Step 2: Consider the Domain of Integration

The integral runs from \(-\infty\) to \(\infty\), which means we need to assess the behavior of the integrand at both ends and through its domain.

Step 3: Determine Convergence at Infinity

The behavior at infinity is determined by the degree of the polynomial in the denominator. As \(x \to \pm\infty\), \(x^4\) dominates, and \(\frac{1}{x^4 + 3x^2 + 2} \approx \frac{1}{x^4}\). Since \(\int \frac{1}{x^4} \ dx\) converges as \(x \to \infty\), this suggests the given integral converges in these regions.

Step 4: Determine Convergence at Zero

Near zero, the polynomial \(x^4 + 3x^2 + 2\) does not change sign and remains positive. Therefore, this does not cause an issue with convergence, as it does not lead to a division by zero.

Step 5: Split the Integral for Evaluation

Since we already suspect convergence, split the integral: \( \int_{-\infty}^{0} \frac{1}{x^4 + 3x^2 + 2} \, dx + \int_{0}^{\infty} \frac{1}{x^4 + 3x^2 + 2} \, dx \). The symmetry of the function due to even power terms suggests both integrals are equal.

Step 6: Simplify the Integral

Perform a partial fraction decomposing: \( \frac{1}{(x^2+1)(x^2+2)} = \frac{A}{x^2+1} + \frac{B}{x^2+2} \). Solving this gives \( A = 1 \) and \( B = -1 \). This simplifies the integral into two easier integrals.

Step 7: Evaluate Each Simplified Integral

These integrals: \( \int \frac{1}{x^2+1} \, dx = \arctan(x) \) and \( \int \frac{1}{x^2+2} \, dx = \frac{1}{\sqrt{2}} \arctan\left(\frac{x}{\sqrt{2}}\right) \) are classic and converge as shown by their limits.

Step 8: Calculate Final Value

Evaluate both integrals from \(-\infty\) to \(\infty\). Using limits, \( \arctan(x) \to [-\frac{\pi}{2}, \frac{\pi}{2}] \) and similar for the other, giving the combined result \( \frac{\pi}{2} \). Thus, the original integral evaluates to \( \frac{\pi}{2} \).

Key Concepts

Convergence and Divergence

When dealing with improper integrals, one of the first tasks is to determine whether the integral converges or diverges. **Convergence** implies that the integral approaches a finite limit as the bounds extend to infinity (or minus infinity), whereas **divergence** indicates that the integral does not settle at a finite value. This is crucial when handling integrals with infinite limits or unbounded integrands.

For the integral \[\int_{-\infty}^{\infty} \frac{1}{x^{4}+3 x^{2}+2} \, dx\]to converge, we need the expression under the integral sign to approach zero sufficiently quickly as \(x\) becomes very large or very small. In our case, the polynomial \(x^4 + 3x^2 + 2\) can be factored to \((x^2 + 1)(x^2 + 2)\). At infinite bounds, the leading term, \(x^4\), in the denominator dominates, suggesting that the integrand behaves like \(\frac{1}{x^4}\), which is known to converge when integrated over infinite bounds. This analysis at both ends coupled with the behavior around zero indicates that the improper integral converges.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify rational expressions, making them easier to integrate. It involves breaking down a complex fraction into a sum of simpler fractions whose denominators are factors of the original denominator.

In accomplishing this, consider:\[ \frac{1}{(x^2 + 1)(x^2 + 2)} \]as the integrand of our improper integral. The task is to express this fraction as a sum of simpler fractions: \[ \frac{A}{x^2 + 1} + \frac{B}{x^2 + 2}.\]By solving for \(A\) and \(B\), we obtain \(A = 1\) and \(B = -1\) respectively. This technique simplifies the integration process because it separates the problem into manageable pieces. Each part corresponds to a standard integral formula, facilitating easier evaluation of the original integral.

Definite Integrals

A definite integral calculates the net area under a curve over a specific interval. It is denoted by the integral sign with upper and lower limits, capturing the concept of accumulating quantities from one point to another.

In this problem, definite integrals are considered over infinite intervals: \[ \int_{-\infty}^{0} \frac{1}{x^4 + 3x^2 + 2} \, dx + \int_{0}^{\infty} \frac{1}{x^4 + 3x^2 + 2} \, dx,\]which are split because of the symmetry in the function's behavior. The evaluation was simplified using partial fractions into more standard forms: \[ \int \frac{1}{x^2+1} \, dx = \arctan(x)\] and \[ \int \frac{1}{x^2+2} \, dx = \frac{1}{\sqrt{2}} \arctan\left(\frac{x}{\sqrt{2}}\right).\]Both these indefinite integrals translate into finite results as limits approach infinity or zero. Evaluating from \(-\infty\) to \(\infty\), results in the original improper integral yielding a final value of \(\frac{\pi}{2}\), affirming convergence.