`int x^3 log x dx`

Answer

To solve the integral \( \int x^3 \log x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)
Let:
- \( u = \log x \) (which will simplify when differentiated)
- \( dv = x^3 \, dx \) (which can be easily integrated)

### Step 2: Differentiate \( u \) and Integrate \( dv \)
Now we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = \frac{1}{x} \, dx \]
- Integrate \( dv \):
\[ v = \int x^3 \, dx = \frac{x^4}{4} \]

### Step 3: Apply Integration by Parts
Now, substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula:
\[ \int x^3 \log x \, dx = uv - \int v \, du \]
This gives us:
\[ \int x^3 \log x \, dx = \log x \cdot \frac{x^4}{4} - \int \frac{x^4}{4} \cdot \frac{1}{x} \, dx \]

### Step 4: Simplify the Integral
The integral simplifies to:
\[ \int x^3 \log x \, dx = \frac{x^4}{4} \log x - \frac{1}{4} \int x^3 \, dx \]

### Step 5: Integrate \( x^3 \)
Now, we need to calculate \( \int x^3 \, dx \):
\[ \int x^3 \, dx = \frac{x^4}{4} \]

### Step 6: Substitute Back
Substituting this back into our equation:
\[ \int x^3 \log x \, dx = \frac{x^4}{4} \log x - \frac{1}{4} \cdot \frac{x^4}{4} + C \]
This simplifies to:
\[ \int x^3 \log x \, dx = \frac{x^4}{4} \log x - \frac{x^4}{16} + C \]

### Final Answer
Thus, the final answer is:
\[ \int x^3 \log x \, dx = \frac{x^4}{4} \log x - \frac{x^4}{16} + C \]