Evaluate `int((x+1)/(x))(x+logx)^(2)dx`

Explanation

To evaluate the integral

\[ \int \frac{x+1}{x} (x + \log x)^2 \, dx, \]

we can follow these steps:

### Step 1: Simplify the integrand
First, we rewrite the term \(\frac{x+1}{x}\):

\[ \frac{x+1}{x} = 1 + \frac{1}{x}. \]

So, we can express the integral as:

\[ \int \left(1 + \frac{1}{x}\right) (x + \log x)^2 \, dx. \]

### Step 2: Use substitution
Next, we will use the substitution \(t = x + \log x\). To find \(dt\), we differentiate \(t\):

\[ dt = \left(1 + \frac{1}{x}\right) dx. \]

This implies that:

\[ \left(1 + \frac{1}{x}\right) dx = dt. \]

### Step 3: Rewrite the integral
Now we can rewrite the integral in terms of \(t\):

\[ \int (x + \log x)^2 \left(1 + \frac{1}{x}\right) dx = \int t^2 \, dt. \]

### Step 4: Integrate
Now we can integrate \(t^2\):

\[ \int t^2 \, dt = \frac{t^3}{3} + C. \]

### Step 5: Substitute back
Finally, we substitute back \(t = x + \log x\):

\[ \frac{(x + \log x)^3}{3} + C. \]

### Final Answer
Thus, the evaluated integral is:

\[ \int \frac{x+1}{x} (x + \log x)^2 \, dx = \frac{(x + \log x)^3}{3} + C. \]

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