Question: What is the integration of log x?
- x log x – x + C
- x log x + C
- log x + C
- None of the above
Answer: (A) x log x – x + C
Integration of log x Solution:
This method involves choosing two functions, u and v, and then using the following formula:
∫ u(x) v′(x) dx = u(x) v(x) - ∫ u′(x) v(x) dx
To integrate log x, we can choose u(x) = log x and v′(x) = 1. This gives us:
∫ log x dx = x log x - ∫ 1 / x dx
The second integral can be evaluated using the following formula:
∫ 1 / x dx = ln |x| + C
Therefore, the integration of log x is:
∫ log x dx = x log x - ln |x| + C
We can simplify this expression by combining the two constants into one, giving us:
∫ log x dx = x log x - x + C
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