Choose the correct answer of the given question
`int (1-x^4)/(1-x) dx = ___ +c`

Options

A. `x^2/2+x^3/3`
B. `x+x^2/2+x^3/3+x^4/4`
C. `x^2/2+x^3/3+x^4/4`
D. `x^2/2+x^4/4`

Answer

To solve the integral \( \int \frac{1 - x^4}{1 - x} \, dx \), we can start by simplifying the integrand.

### Step 1: Simplify the integrand
We can rewrite the integrand by performing polynomial long division or by factoring. Notice that \( 1 - x^4 \) can be factored as:
\[ 1 - x^4 = (1 - x)(1 + x + x^2 + x^3) \]
Thus, we can rewrite the integral as:
\[ \int \frac{(1 - x)(1 + x + x^2 + x^3)}{1 - x} \, dx \]
The \( (1 - x) \) in the numerator and denominator cancels out, leaving us with:
\[ \int (1 + x + x^2 + x^3) \, dx \]

### Step 2: Integrate each term
Now we can integrate each term separately:
\[ \int (1 + x + x^2 + x^3) \, dx = \int 1 \, dx + \int x \, dx + \int x^2 \, dx + \int x^3 \, dx \]
Calculating each integral:
- \( \int 1 \, dx = x \)
- \( \int x \, dx = \frac{x^2}{2} \)
- \( \int x^2 \, dx = \frac{x^3}{3} \)
- \( \int x^3 \, dx = \frac{x^4}{4} \)

### Step 3: Combine the results
Combining all these results, we have:
\[ \int (1 + x + x^2 + x^3) \, dx = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + C \]

### Final Answer
Thus, the integral \( \int \frac{1 - x^4}{1 - x} \, dx \) evaluates to:
\[ x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + C \]