The integral `int(2x^(3)-1)/(x^(4)+x)dx` is equal to (here C is a constant of intergration)

Options

A. `(1)/(2)"log"_(e)(|x^(3)+1|)/(x^(2))+C`
B. `(1)/(2)"log"_(e)(|x^(3)+1|^(2))/(|x^(3)|)+C`
C. `"log"_(e)|(x^(3)+1)/(x)|+C`
D. `"log"_(e)(|x^(3)+1|)/(x^(2))+C`

Explanation

To solve the integral \(\int \frac{2x^3 - 1}{x^4 + x} \, dx\), we can follow these steps:

### Step 1: Simplify the Integral
We start with the integral:
\[ \int \frac{2x^3 - 1}{x^4 + x} \, dx \]
We can factor out \(x\) from the denominator:
\[ x^4 + x = x(x^3 + 1) \]
Thus, we rewrite the integral as:
\[ \int \frac{2x^3 - 1}{x(x^3 + 1)} \, dx \]

### Step 2: Perform Polynomial Long Division
Since the degree of the numerator is equal to the degree of the denominator, we can perform polynomial long division:
1. Divide \(2x^3\) by \(x^3\) to get \(2\).
2. Multiply \(2\) by \(x^3 + 1\) to get \(2x^3 + 2\).
3. Subtract this from \(2x^3 - 1\):
\[ (2x^3 - 1) - (2x^3 + 2) = -3 \]
So, we can express the integral as:
\[ \int \left(2 + \frac{-3}{x(x^3 + 1)}\right) \, dx \]

### Step 3: Split the Integral
Now we can split the integral into two parts:
\[ \int 2 \, dx - 3 \int \frac{1}{x(x^3 + 1)} \, dx \]

### Step 4: Integrate the First Part
The first integral is straightforward:
\[ \int 2 \, dx = 2x \]

### Step 5: Integrate the Second Part
For the second integral, we can use partial fraction decomposition:
\[ \frac{1}{x(x^3 + 1)} = \frac{A}{x} + \frac{Bx^2 + Cx + D}{x^3 + 1} \]
Multiplying through by the denominator \(x(x^3 + 1)\) and equating coefficients will allow us to solve for \(A\), \(B\), \(C\), and \(D\).

After finding the coefficients, we can integrate each term separately. The integral of \(\frac{1}{x}\) is \(\ln|x|\), and the integral of \(\frac{1}{x^3 + 1}\) can be solved using substitution or recognizing it as a standard integral.

### Step 6: Combine the Results
Finally, we combine the results of both integrals and add the constant of integration \(C\):
\[ \int \frac{2x^3 - 1}{x^4 + x} \, dx = 2x - 3\left(\text{result from the second integral}\right) + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{2x^3 - 1}{x^4 + x} \, dx = 2x - 3\left(\text{result from the second integral}\right) + C \]