`int(x^(4))/(x^(2)+1)dx` is equal to

Options

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

Answer

To solve the integral \(\int \frac{x^4}{x^2 + 1} \, dx\), we will follow these steps:

### Step 1: Rewrite the integrand
We can rewrite \(x^4\) in a way that makes the integration easier. We can express \(x^4\) as:
\[ x^4 = (x^2 + 1)(x^2 - 1) + (x^2 + 1) \]
This gives us:
\[ \frac{x^4}{x^2 + 1} = x^2 - 1 + 1 \]
So, we can rewrite the integral as:
\[ \int \frac{x^4}{x^2 + 1} \, dx = \int (x^2 - 1 + 1) \, dx \]

### Step 2: Split the integral
Now, we can split the integral into three separate integrals:
\[ \int (x^2 - 1 + 1) \, dx = \int x^2 \, dx - \int 1 \, dx + \int 1 \, dx \]

### Step 3: Integrate each term
Now, we will integrate each term separately.

1. For \(\int x^2 \, dx\):
\[ \int x^2 \, dx = \frac{x^3}{3} \]

2. For \(-\int 1 \, dx\):
\[ -\int 1 \, dx = -x \]

3. For \(\int 1 \, dx\):
\[ \int 1 \, dx = x \]

### Step 4: Combine the results
Now, we combine the results of the integrals:
\[ \int \frac{x^4}{x^2 + 1} \, dx = \frac{x^3}{3} - x + x + C \]
The \(-x\) and \(+x\) cancel each other out, so we have:
\[ \int \frac{x^4}{x^2 + 1} \, dx = \frac{x^3}{3} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{x^4}{x^2 + 1} \, dx = \frac{x^3}{3} + C \]