The value of integral `underset(2)overset(4)int(dx)/(x)` is :-

Options

A. `3 log _(e)2`
B. `log _(e)2`
C. `log _(e)4`
D. `2 log _(e) 8`

Explanation

To solve the integral \(\int_{2}^{4} \frac{dx}{x}\), we will follow these steps:

### Step 1: Set Up the Integral
We start by defining the integral:
\[ I = \int_{2}^{4} \frac{dx}{x} \]

### Step 2: Find the Antiderivative
The antiderivative of \(\frac{1}{x}\) is \(\ln |x|\). Therefore, we can write:
\[ I = \left[ \ln |x| \right]_{2}^{4} \]

### Step 3: Evaluate the Integral
Now we evaluate the definite integral from 2 to 4:
\[ I = \ln |4| - \ln |2| \]
Since both 4 and 2 are positive, we can drop the absolute value:
\[ I = \ln 4 - \ln 2 \]

### Step 4: Use Logarithmic Properties
Using the property of logarithms that states \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\), we can simplify:
\[ I = \ln \left(\frac{4}{2}\right) = \ln 2 \]

### Final Result
Thus, the value of the integral is:
\[ I = \ln 2 \]