`int(logx)/(x)dx=?`

Options

A. `(1)/(2)(logx)^(2)+C`
B. `-(1)/(2)(logx)^(2)+C`
C. `(2)/(x^(2))+C`
D. `(-2)/(x^(2))+C`

Explanation

To solve the integral \(\int \frac{\log x}{x} \, dx\), we can use the substitution method. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \log x \). Then, we differentiate both sides to find \( dx \):
\[ dt = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, dt \]
Since \( x = e^t \) (from \( t = \log x \)), we can substitute \( dx \) in terms of \( t \):
\[ dx = e^t \, dt \]

### Step 2: Rewrite the Integral
Now, substituting \( t \) and \( dx \) into the integral:
\[ \int \frac{\log x}{x} \, dx = \int t \cdot \frac{1}{x} \cdot dx = \int t \cdot dt \]

### Step 3: Integrate
Now we can integrate \( t \):
\[ \int t \, dt = \frac{t^2}{2} + C \]

### Step 4: Back Substitute
Now, we substitute back \( t = \log x \):
\[ \frac{t^2}{2} + C = \frac{(\log x)^2}{2} + C \]

### Final Answer
Thus, the integral \(\int \frac{\log x}{x} \, dx\) is:
\[ \int \frac{\log x}{x} \, dx = \frac{(\log x)^2}{2} + C \]

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