`int((log _(e)x)^(3))/(x)dx`

Answer

To solve the integral \( I = \int \frac{(\log_e x)^3}{x} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \log_e x \). Then, differentiate both sides with respect to \( x \):
\[ \frac{dt}{dx} = \frac{1}{x} \implies dx = x \, dt \]
Since \( x = e^t \), we can also express \( dx \) in terms of \( t \):
\[ dx = e^t \, dt \]

### Step 2: Rewrite the Integral
Now substitute \( t \) into the integral:
\[ I = \int \frac{t^3}{x} \, dx = \int t^3 \cdot \frac{dx}{x} \]
Since \( \frac{dx}{x} = dt \), we have:
\[ I = \int t^3 \, dt \]

### Step 3: Integrate
Now, we can integrate \( t^3 \):
\[ I = \frac{t^{3+1}}{3+1} + C = \frac{t^4}{4} + C \]

### Step 4: Resubstitute
Now, substitute back \( t = \log_e x \):
\[ I = \frac{(\log_e x)^4}{4} + C \]

### Final Answer
Thus, the final answer is:
\[ I = \frac{(\log_e x)^4}{4} + C \]

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