**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:

The integration of log x is a fundamental concept in calculus. To integrate, we use integration by parts, where u = log x and dv = dx. The result is x log x − x + C, where C is the constant of integration, highlighting the relationship between logarithmic and polynomial functions.

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
```

```
∫ 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`

## How Many Centimeters is an Inch

An inch is a common unit of measurement in the United States, often used in everyday contexts like measuring height or width. To convert inches to centimeters, it’s essential to know the conversion factor. One inch is equal to 2.54 centimeters. This means that if you want to find out how many centimeters are in several inches, simply multiply the number of inches by 2.54. For example, if you have 5 inches, multiplying by 2.54 gives you 12.7 centimeters. Understanding how many centimeters is an inch helps make measurements more precise, especially in fields like engineering and design.