CA Foundation · Quantitative Aptitude · Differential and Integral Calculus
The value of ∫₁ᵉ log x dx (natural logarithm) is:
The value is 1. Integration by parts gives the antiderivative x log x − x. At x = e it equals 0, and at x = 1 it equals −1. Subtracting the lower-limit value from the upper-limit value gives 0 − (−1) = 1.
- A1Correct
- Be − 1
- Ce
- D0
Explanation
Write log x as 1·log x and use parts with u = log x, dv = dx. The integral is x log x − x. At e this is e − e = 0. At 1 it is 0 − 1 = −1. The difference is 0 − (−1) = 1. The option e − 1 results from forgetting to evaluate the lower limit correctly.
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