`intx^3(logx)^2dx `

Options

A. `1/32 x^4[8(log x)^2+4log x -1 ]`
B. `1/32 x^4[8(log x)^2+4log x +1 ]`
C. `1/32 x^4[8(log x)^2-4log x +1 ]`
D. None of these

Answer

To solve the integral \( \int x^3 (\log x)^2 \, dx \), we will use the method of integration by parts. The integration by parts formula is given by:

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

### Step 1: Choose \( u \) and \( dv \)

Let:
- \( u = (\log x)^2 \) (First function)
- \( dv = x^3 \, dx \) (Second function)

### Step 2: Differentiate \( u \) and Integrate \( dv \)

Now we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = 2 \log x \cdot \frac{1}{x} \, dx = \frac{2 \log x}{x} \, dx \]

- Integrate \( dv \):
\[ v = \int x^3 \, dx = \frac{x^4}{4} \]

### Step 3: Apply Integration by Parts

Now we apply the integration by parts formula:
\[ \int x^3 (\log x)^2 \, dx = uv - \int v \, du \]
Substituting the values we found:
\[ = \left(\frac{x^4}{4} (\log x)^2\right) - \int \left(\frac{x^4}{4} \cdot \frac{2 \log x}{x}\right) \, dx \]
This simplifies to:
\[ = \frac{x^4}{4} (\log x)^2 - \frac{1}{2} \int x^3 \log x \, dx \]

### Step 4: Solve \( \int x^3 \log x \, dx \) Using Integration by Parts Again

For the integral \( \int x^3 \log x \, dx \), we apply integration by parts again:
Let:
- \( u = \log x \)
- \( dv = x^3 \, dx \)

Then:
- \( du = \frac{1}{x} \, dx \)
- \( v = \frac{x^4}{4} \)

Applying integration by parts:
\[ \int x^3 \log x \, dx = \left(\frac{x^4}{4} \log x\right) - \int \left(\frac{x^4}{4} \cdot \frac{1}{x}\right) \, dx \]
This simplifies to:
\[ = \frac{x^4}{4} \log x - \frac{1}{4} \int x^3 \, dx \]
Calculating \( \int x^3 \, dx \):
\[ = \frac{x^4}{4} \]
Thus:
\[ \int x^3 \log x \, dx = \frac{x^4}{4} \log x - \frac{1}{4} \cdot \frac{x^4}{4} = \frac{x^4}{4} \log x - \frac{x^4}{16} \]

### Step 5: Substitute Back

Now substitute back into our earlier expression:
\[ \int x^3 (\log x)^2 \, dx = \frac{x^4}{4} (\log x)^2 - \frac{1}{2} \left( \frac{x^4}{4} \log x - \frac{x^4}{16} \right) \]
This simplifies to:
\[ = \frac{x^4}{4} (\log x)^2 - \frac{x^4}{8} \log x + \frac{x^4}{32} \]

### Step 6: Combine Terms

Now, we can factor out \( \frac{x^4}{32} \):
\[ = \frac{x^4}{32} \left( 8 (\log x)^2 - 4 \log x + 1 \right) + C \]

### Final Answer

Thus, the final result is:
\[ \int x^3 (\log x)^2 \, dx = \frac{x^4}{32} \left( 8 (\log x)^2 - 4 \log x + 1 \right) + C \]