The value of the integral is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: B
Solution
Split the integral at x=1 where ln x changes sign.
For x in [e⁻¹, 1]: ln x ≤ 0, so |ln x / x| = −(ln x)/x
For x in [1, e²]: ln x ≥ 0, so |ln x / x| = (ln x)/x
Using ∫(ln x)/x dx = (ln x)²/2:
First part: −[(ln x)²/2] from e⁻¹ to 1 = −[0 − (−1)²/2] = −[−1/2] = 1/2
Second part: [(ln x)²/2] from 1 to e² = [(2)²/2 − 0] = 2
Total = 1/2 + 2 = 5/2
For x in [e⁻¹, 1]: ln x ≤ 0, so |ln x / x| = −(ln x)/x
For x in [1, e²]: ln x ≥ 0, so |ln x / x| = (ln x)/x
Using ∫(ln x)/x dx = (ln x)²/2:
First part: −[(ln x)²/2] from e⁻¹ to 1 = −[0 − (−1)²/2] = −[−1/2] = 1/2
Second part: [(ln x)²/2] from 1 to e² = [(2)²/2 − 0] = 2
Total = 1/2 + 2 = 5/2
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