**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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