Question

Select the integrals below that are convergent.

1) ∫(1/x) dx from 1 to [infinity]
2) ∫(1/x²) dx from 1 to [infinity]
3) ∫(1/x³) dx from 1 to [infinity]
4) ∫(1/x⁴) dx from 1 to [infinity]

Answer 1

The student is inquiring about the convergence of four integrals. The first integral is divergent, while integrals two, three, and four are convergent because their functions approach zero as x approaches infinity.

Step-by-step explanation:

The student is asking which of the given integrals are convergent when evaluated from 1 to infinity. To determine convergence, we examine the behavior of the function as x approaches infinity. For integral 1 (∫(1/x) dx from 1 to infinity), this integral is divergent because the integral of 1/x is ln(x) which approaches infinity as x approaches infinity. For integral 2 (∫(1/x²) dx from 1 to infinity), the integral is convergent because the integral of 1/x² is -1/x, which approaches 0 as x approaches infinity. Similarly, integrals 3 and 4 (∫(1/x³) dx and ∫(1/x´) dx from 1 to infinity) are both convergent for the same reason.