`intx^3tan^(-1)xdx=`

Options

A. `[(x^4-1)tan^(-1)x-x^3/3+x]`
B. `1/2[(x^4-1)tan^(-1)x-x^3/3+x]`
C. `1/3[(x^4-1)tan^(-1)x-x^3/3+x]`
D. `1/4[(x^4-1)tan^(-1)x-x^3/3+x]`

Answer

To solve the integral \( \int x^3 \tan^{-1}(x) \, dx \), 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:
- \( u = \tan^{-1}(x) \) (which we will differentiate)
- \( dv = x^3 \, dx \) (which we will integrate)

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

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

### Step 3: Apply the integration by parts formula

Now, substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula:
\[ \int x^3 \tan^{-1}(x) \, dx = uv - \int v \, du \]
Substituting the values we found:
\[ = \tan^{-1}(x) \cdot \frac{x^4}{4} - \int \frac{x^4}{4} \cdot \frac{1}{1+x^2} \, dx \]

### Step 4: Simplify the integral

Now we need to simplify the integral:
\[ = \frac{x^4}{4} \tan^{-1}(x) - \frac{1}{4} \int \frac{x^4}{1+x^2} \, dx \]
We can simplify \( \frac{x^4}{1+x^2} \):
\[ \frac{x^4}{1+x^2} = x^2 - \frac{x^2}{1+x^2} \]
So, we rewrite the integral:
\[ \int \frac{x^4}{1+x^2} \, dx = \int x^2 \, dx - \int \frac{x^2}{1+x^2} \, dx \]

### Step 5: Calculate the integrals

Now we calculate each integral:
1. \( \int x^2 \, dx = \frac{x^3}{3} \)
2. For \( \int \frac{x^2}{1+x^2} \, dx \), we can use substitution:
Let \( t = 1 + x^2 \), then \( dt = 2x \, dx \) or \( dx = \frac{dt}{2x} = \frac{dt}{2\sqrt{t-1}} \).
The integral becomes:
\[ \int \frac{x^2}{1+x^2} \, dx = \int \frac{t-1}{t} \cdot \frac{dt}{2\sqrt{t-1}} = \frac{1}{2} \int \left(1 - \frac{1}{t}\right) \sqrt{t-1} \, dt \]

### Step 6: Combine all parts

Now combine everything back:
\[ \int x^3 \tan^{-1}(x) \, dx = \frac{x^4}{4} \tan^{-1}(x) - \frac{1}{4} \left( \frac{x^3}{3} - \text{(result of the second integral)} \right) + C \]

### Final Result

After calculating the second integral and simplifying, we can write the final result of the integral.