`intx^(3) log x dx` is equal to A) `(x^(4) log x )/( 4) + C` B) `(x^(4))/( 8) ( log x - ( 4)/( x^(2)))+C` C) `(x^(4))/( 16) ( 4 log x -1) +C` D) `(x^(4))/( 16) ( 4 log x +1) + C`

Options

A. `(x^(4) log x )/( 4) + C`
B. `(x^(4))/( 8) ( log x - ( 4)/( x^(2)))+C`
C. `(x^(4))/( 16) ( 4 log x -1) +C`
D. `(x^(4))/( 16) ( 4 log x +1) + C`

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 we will differentiate)
- \( dv = x^3 \, dx \) (which we will integrate)

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

### Step 3: Apply the Integration by Parts Formula
Now we apply the integration by parts formula:
\[ \int x^3 \log x \, dx = uv - \int v \, du \]
Substituting \( u \), \( du \), \( v \):
\[ = \log x \cdot \frac{x^4}{4} - \int \frac{x^4}{4} \cdot \frac{1}{x} \, dx \]
This simplifies to:
\[ = \frac{x^4 \log x}{4} - \frac{1}{4} \int x^3 \, dx \]

### Step 4: Compute the Remaining Integral
Now we compute the integral \( \int x^3 \, dx \):
\[ \int x^3 \, dx = \frac{x^4}{4} \]
Substituting this back into our equation:
\[ = \frac{x^4 \log x}{4} - \frac{1}{4} \cdot \frac{x^4}{4} \]
This simplifies to:
\[ = \frac{x^4 \log x}{4} - \frac{x^4}{16} \]

### Step 5: Combine the Terms
Now, we can combine the terms:
\[ = \frac{x^4}{16} \left( 4 \log x - 1 \right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int x^3 \log x \, dx = \frac{x^4}{16} (4 \log x - 1) + C \]