`int(x^(3)-1)^(1//3)x^(5)dx` is equal to

Options

A. `(1)/(7)(x^(3)+1)^(1//3)+(1)/(4)(x^(3)-1)^(3//4)+C`
B. `(1)/(7)(x^(3)-1)^(7//3)+(1)/(4)(x^(3)-1)^(4//3)+C`
C. `(3)/(7)(x^(3)-1)^(7//3)+(1)/(4)(x^(3)-1)^(4//3)+C`
D. None of the above

Explanation

To solve the integral \( \int (x^3 - 1)^{1/3} x^5 \, dx \), we will use substitution and integration techniques. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = x^3 - 1 \). Then, we differentiate to find \( dt \):
\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]

### Step 2: Express \( x^5 \) in terms of \( t \)
From our substitution, we have:
\[ x^3 = t + 1 \quad \Rightarrow \quad x = (t + 1)^{1/3} \]
Thus,
\[ x^5 = (t + 1)^{5/3} \]

### Step 3: Substitute into the integral
Now we substitute \( t \) and \( dx \) into the integral:
\[ \int (x^3 - 1)^{1/3} x^5 \, dx = \int t^{1/3} (t + 1)^{5/3} \cdot \frac{dt}{3x^2} \]
We need to express \( x^2 \) in terms of \( t \):
\[ x^2 = (t + 1)^{2/3} \]
So,
\[ dx = \frac{dt}{3(t + 1)^{2/3}} \]

### Step 4: Rewrite the integral
Now we can rewrite the integral:
\[ \int t^{1/3} (t + 1)^{5/3} \cdot \frac{dt}{3(t + 1)^{2/3}} = \frac{1}{3} \int t^{1/3} (t + 1)^{5/3 - 2/3} \, dt = \frac{1}{3} \int t^{1/3} (t + 1)^{3/3} \, dt \]
This simplifies to:
\[ \frac{1}{3} \int t^{1/3} (t + 1) \, dt \]

### Step 5: Expand and integrate
Now we expand the integrand:
\[ \frac{1}{3} \int (t^{1/3} t + t^{1/3}) \, dt = \frac{1}{3} \int (t^{4/3} + t^{1/3}) \, dt \]
Now we can integrate term by term:
\[ \frac{1}{3} \left( \frac{t^{4/3 + 1}}{4/3 + 1} + \frac{t^{1/3 + 1}}{1/3 + 1} \right) + C = \frac{1}{3} \left( \frac{t^{7/3}}{7/3} + \frac{t^{4/3}}{4/3} \right) + C \]
This simplifies to:
\[ \frac{1}{3} \left( \frac{3}{7} t^{7/3} + \frac{3}{4} t^{4/3} \right) + C = \frac{1}{7} t^{7/3} + \frac{1}{4} t^{4/3} + C \]

### Step 6: Substitute back for \( t \)
Now we substitute back \( t = x^3 - 1 \):
\[ = \frac{1}{7} (x^3 - 1)^{7/3} + \frac{1}{4} (x^3 - 1)^{4/3} + C \]

### Final Answer
Thus, the final answer is:
\[ \int (x^3 - 1)^{1/3} x^5 \, dx = \frac{1}{7} (x^3 - 1)^{7/3} + \frac{1}{4} (x^3 - 1)^{4/3} + C \]