`int(1)/(Kx)dx`=_________ .

Options

A. `log|x|+C`
B. `log|Kx|+C`
C. `1/K log |x|+C`
D. `(-1)/(Kx^2)+C`

Answer

To solve the integral \(\int \frac{1}{Kx} \, dx\), we can follow these steps:

### Step 1: Factor out the constant
The integral can be rewritten by factoring out the constant \(K\):
\[ \int \frac{1}{Kx} \, dx = \frac{1}{K} \int \frac{1}{x} \, dx \]

### Step 2: Solve the integral of \(\frac{1}{x}\)
The integral of \(\frac{1}{x}\) is a well-known result:
\[ \int \frac{1}{x} \, dx = \ln |x| + C \]
where \(C\) is the constant of integration.

### Step 3: Substitute back
Now, substitute this result back into the equation:
\[ \frac{1}{K} \int \frac{1}{x} \, dx = \frac{1}{K} (\ln |x| + C) \]
This simplifies to:
\[ \frac{1}{K} \ln |x| + \frac{C}{K} \]

### Step 4: Write the final answer
Since \(\frac{C}{K}\) is still a constant, we can denote it as a new constant \(C'\):
\[ \int \frac{1}{Kx} \, dx = \frac{1}{K} \ln |x| + C' \]

Thus, the final answer is:
\[ \int \frac{1}{Kx} \, dx = \frac{1}{K} \ln |x| + C \]