Integration of log x – The Logarithm of x Integral Explained

Question: What is the integration of log x?

  1. x log x – x + C
  2. x log x + C
  3. log x + C
  4. 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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