82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

Verified step by step guidance

1. Identify the type of integral: This is an improper integral with limits from \(-\infty\) to \(\infty\), so we need to consider the behavior of the integrand as \(x\) approaches \(\pm \infty\).
2. Examine the integrand \(f(x) = \frac{x^3}{1 + x^8}\) for symmetry: Determine if the function is even, odd, or neither by checking \(f(-x)\).
3. Since \(f(-x) = \frac{(-x)^3}{1 + (-x)^8} = \frac{-x^3}{1 + x^8} = -f(x)\), the function is odd.
4. Recall that the integral of an odd function over symmetric limits \([-a, a]\) is zero, provided the integral converges.
5. Check the convergence of the integral by analyzing the behavior of \(f(x)\) as \(x \to \infty\): Since the degree of the denominator is higher than the numerator, the integrand approaches zero fast enough to ensure convergence.

Key Concepts

Improper Integrals

Improper integrals involve integration over infinite intervals or integrands with infinite discontinuities. To evaluate them, limits are used to define the integral as a limit of definite integrals over finite intervals. Determining convergence or divergence is essential before finding a value.

Even and Odd Functions

A function is odd if f(-x) = -f(x) and even if f(-x) = f(x). For integrals over symmetric intervals [-a, a], the integral of an odd function is zero, while the integral of an even function may be nonzero. Recognizing function symmetry simplifies evaluation.

Behavior of Rational Functions at Infinity

Rational functions are ratios of polynomials. Their behavior as x approaches infinity depends on the degrees of numerator and denominator. If the denominator grows faster, the function approaches zero, which affects the convergence of improper integrals over infinite intervals.