`int_2^4 1/x dx` is equal to

Options

A. `In 2`
B. `2 In 2`
C. `In(3/2)`
D. `2 In 4`

Answer

To solve the integral \( \int_2^4 \frac{1}{x} \, dx \), we can follow these steps:

### Step 1: Identify the integral
We need to evaluate the integral of the function \( \frac{1}{x} \) from 2 to 4.

### Step 2: Use the integration formula
The integral of \( \frac{1}{x} \) is given by:
\[ \int \frac{1}{x} \, dx = \ln |x| + C \]
where \( C \) is the constant of integration.

### Step 3: Apply the limits of integration
Now, we will evaluate the definite integral from 2 to 4:
\[ \int_2^4 \frac{1}{x} \, dx = \left[ \ln |x| \right]_2^4 \]

### Step 4: Substitute the limits
Substituting the upper limit (4) and the lower limit (2) into the expression:
\[ = \ln |4| - \ln |2| \]
Since both 4 and 2 are positive, we can drop the absolute value:
\[ = \ln 4 - \ln 2 \]

### Step 5: Use the logarithmic property
We can use the property of logarithms that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \):
\[ = \ln \left( \frac{4}{2} \right) \]

### Step 6: Simplify the fraction
Calculating \( \frac{4}{2} \):
\[ = \ln 2 \]

### Final Answer
Thus, the value of the integral \( \int_2^4 \frac{1}{x} \, dx \) is:
\[ \ln 2 \]

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