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

Options

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

Answer

To solve the integral

\[ \int \frac{2x^3 - 1}{x^4 + x} \, dx, \]

we can break it down into two separate integrals:

\[ \int \frac{2x^3}{x^4 + x} \, dx - \int \frac{1}{x^4 + x} \, dx. \]

### Step 1: Simplifying the First Integral

For the first integral, we can simplify the fraction:

\[ \frac{2x^3}{x^4 + x} = \frac{2x^3}{x(x^3 + 1)} = \frac{2x^2}{x^3 + 1}. \]

Thus, we rewrite the integral as:

\[ \int \frac{2x^2}{x^3 + 1} \, dx. \]

### Step 2: Substitution for the First Integral

Let \( t = x^3 + 1 \). Then, differentiating both sides gives:

\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2}. \]

Substituting \( x^2 = \frac{t - 1}{x} \) into the integral:

\[ \int \frac{2x^2}{t} \cdot \frac{dt}{3x^2} = \frac{2}{3} \int \frac{1}{t} \, dt = \frac{2}{3} \ln |t| + C = \frac{2}{3} \ln |x^3 + 1| + C. \]

### Step 3: Simplifying the Second Integral

Now, we focus on the second integral:

\[ \int \frac{1}{x^4 + x} \, dx. \]

Factoring the denominator gives:

\[ x^4 + x = x(x^3 + 1) = x(x + 1)(x^2 - x + 1). \]

### Step 4: Partial Fraction Decomposition

We can express:

\[ \frac{1}{x(x + 1)(x^2 - x + 1)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{Cx + D}{x^2 - x + 1}. \]

Multiplying through by the denominator and equating coefficients will yield values for \( A, B, C, \) and \( D \).

### Step 5: Integrating Each Term

Once we find \( A, B, C, \) and \( D \), we can integrate each term separately. The integrals will involve logarithmic functions for the linear terms and arctangent or logarithmic forms for the quadratic term.

### Step 6: Combining Results

Finally, we combine the results of both integrals:

\[ \int \frac{2x^3 - 1}{x^4 + x} \, dx = \frac{2}{3} \ln |x^3 + 1| - \left( \text{result of the second integral} \right) + C. \]

### Final Answer

The final result will be:

\[ \int \frac{2x^3 - 1}{x^4 + x} \, dx = \frac{2}{3} \ln |x^3 + 1| - \text{(result from the second integral)} + C. \]