Question

Determine whether the improper integral is convergent or divergent without solving the integral. Use an appropriate inequality to support your conclusion.

[infinity] 2x
∫ ------------------ dx
¹ x² + 3x² + 2

Answer 1

The improper integral in question is divergent, which is demonstrated by comparing it to a simpler integral that diverges using the comparison test.

Step-by-step explanation:

The student has asked to determine whether the improper integral ∞ ∫ 2x/(x² + 3x² + 2) dx, from 1 to infinity, is convergent or divergent without actually solving the integral. To approach this, we can look for a function that is easier to integrate and dominates the given function from some point onwards. Here, the denominator can be simplified to 4x² + 2. We know that for x ≥ 1, 4x² will always be greater than or equal to 4x² + 2, leading to the inequality:

∞ ∫ 2x/(4x² + 2) dx ≤ ∞ ∫ 2x/(4x²) dx

After simplifying the right-hand side, we get:

∞ ∫ 1/(2x) dx

This results in an integral that is a p-integral with p = 1, which is known to diverge. Hence, by the comparison test, since the simplified integral diverges, the original integral must also diverge.