Evaluate the following integrals:
`int (1)/(x (2+3 log x))dx`

Answer

To evaluate the integral \( \int \frac{1}{x(2 + 3 \log x)} \, dx \), we can follow these steps:

### Step 1: Substitution
Let's make a substitution to simplify the integral. We set:
\[ t = 2 + 3 \log x \]
Next, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 3 \cdot \frac{1}{x} \implies dt = \frac{3}{x} \, dx \implies dx = \frac{x}{3} \, dt \]

### Step 2: Express \( x \) in terms of \( t \)
From our substitution, we can express \( \log x \) in terms of \( t \):
\[ \log x = \frac{t - 2}{3} \implies x = e^{\frac{t - 2}{3}} \]

### Step 3: Substitute into the integral
Now we can substitute \( x \) and \( dx \) back into the integral:
\[ \int \frac{1}{x(2 + 3 \log x)} \, dx = \int \frac{1}{x \cdot t} \cdot \frac{x}{3} \, dt \]
The \( x \) cancels out:
\[ = \int \frac{1}{3t} \, dt = \frac{1}{3} \int \frac{1}{t} \, dt \]

### Step 4: Integrate
The integral of \( \frac{1}{t} \) is:
\[ \frac{1}{3} \log |t| + C \]

### Step 5: Substitute back for \( t \)
Now we substitute back \( t = 2 + 3 \log x \):
\[ = \frac{1}{3} \log |2 + 3 \log x| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{x(2 + 3 \log x)} \, dx = \frac{1}{3} \log |2 + 3 \log x| + C \]

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