Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.

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Question: Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.

Options:

Correct Answer: 3

Solution:

The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.

Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.

Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.

Correct Answer: -2/3

Step 1: Identify the integral to be calculated: ∫ from 0 to 1 of (x^2 + 1/x^2) dx.
Step 2: Break down the integral into two parts: ∫ from 0 to 1 of x^2 dx and ∫ from 0 to 1 of 1/x^2 dx.
Step 3: Calculate the first part: ∫ x^2 dx = (x^3)/3.
Step 4: Evaluate the first part from 0 to 1: [(1^3)/3 - (0^3)/3] = (1/3 - 0) = 1/3.
Step 5: Calculate the second part: ∫ 1/x^2 dx = -1/x.
Step 6: Evaluate the second part from 0 to 1: [-1/1 - (-1/0)]. Note: -1/0 is undefined, but we consider the limit as x approaches 0, which goes to -∞.
Step 7: Combine the results: 1/3 + (limit as x approaches 0 of -1/x) = 1/3 - ∞ = -∞.
Step 8: Since the second part diverges, the overall integral diverges.

Definite Integrals – The question tests the ability to evaluate a definite integral over a specified interval.
Integration Techniques – It requires knowledge of how to integrate polynomial and rational functions.
Handling Improper Integrals – The integral has a term (1/x^2) that may lead to issues at x=0, testing the understanding of limits.