Evaluate the improper integral: integral from 1 to infinity of (1 / x^2) dx.
Answer and explanation
Correct answer: A. 1
- Step 1: Write the improper integral as a limit: lim (b -> infinity) [integral from 1 to b of x^(-2) dx].
- Step 2: Find the antiderivative: [-1 / x] from 1 to b = (-1/b) - (-1/1) = 1 - 1/b.
- Step 3: Take the limit as b -> infinity: lim (b -> infinity) (1 - 1/b) = 1 - 0 = 1. The integral converges to 1.
Answer
1.
Why not the others: 'infinity (diverges)' misidentifies this convergent p-series integral with p = 2 > 1; '1/2' makes an integration coefficient error; '0' evaluates the upper limit term alone.
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