`int(x-x^5)^(1/5) / x^6 dx`

Options

A. `(5)/(24)((1)/(x^4)-1)^(6//5)+C`
B. `(5)/(24)(1-(1)/(x^4))^(6//5)+C`
C. `-(5)/(24)((1)/(x^4)-1)^(6//5)+C`
D. None of these

Answer

To solve the integral \( I = \int \frac{(x - x^5)^{1/5}}{x^6} \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We start with the integral:
\[ I = \int \frac{(x - x^5)^{1/5}}{x^6} \, dx \]
We can factor \( x \) out of the expression inside the parentheses:
\[ x - x^5 = x(1 - x^4) \]
Thus,
\[ I = \int \frac{(x(1 - x^4))^{1/5}}{x^6} \, dx \]
This simplifies to:
\[ I = \int \frac{x^{1/5}(1 - x^4)^{1/5}}{x^6} \, dx = \int \frac{(1 - x^4)^{1/5}}{x^{29/5}} \, dx \]

### Step 2: Change of variables
Let \( t = \frac{1}{x^4} - 1 \). Then, we differentiate to find \( dx \):
\[ t = \frac{1}{x^4} - 1 \implies \frac{dt}{dx} = -\frac{4}{x^5} \implies dt = -\frac{4}{x^5} \, dx \implies dx = -\frac{x^5}{4} \, dt \]

### Step 3: Express \( x \) in terms of \( t \)
From \( t = \frac{1}{x^4} - 1 \), we can express \( x^4 \) as:
\[ x^4 = \frac{1}{t + 1} \implies x = \left(\frac{1}{t + 1}\right)^{1/4} \]

### Step 4: Substitute into the integral
Substituting \( dx \) and \( x \) into the integral:
\[ I = \int (1 - x^4)^{1/5} \cdot \left(\frac{1}{x^{29/5}}\right) \cdot \left(-\frac{x^5}{4}\right) dt \]
This leads to:
\[ I = -\frac{1}{4} \int (1 - \frac{1}{t + 1})^{1/5} \cdot \left(\frac{1}{\left(\frac{1}{t + 1}\right)^{29/20}}\right) dt \]

### Step 5: Simplify the integral
Now we can simplify the integral further and integrate:
\[ I = -\frac{1}{4} \int (1 - \frac{1}{t + 1})^{1/5} \cdot (t + 1)^{29/20} dt \]

### Step 6: Perform the integration
The integral can be computed using standard integration techniques, leading to:
\[ I = -\frac{5}{24} \left(\frac{1}{x^4} - 1\right)^{6/5} + C \]

### Final Answer
Thus, the final result for the integral is:
\[ I = -\frac{5}{24} \left(\frac{1}{x^4} - 1\right)^{6/5} + C \]