The value of `int((x-x^(3))^(1//3))/(x^(4))dx` is

Options

A. `(3)/(8)((1)/(x^(2))-1)^(4//3)+C`
B. `-(3)/(8)((1)/(x^(2))-1)^(4//3)+C`
C. `(1)/(8)(1-(1)/(x^(2)))^(4//3)+1`
D. none of these

Answer

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

### Step 1: Simplify the integrand
We start by factoring out \( x^3 \) from the expression inside the integral:

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

Thus, we can rewrite the integral as:

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

### Step 2: Rewrite the integral
Now, we can express this as:

\[ \int \frac{x^{1/3}(1 - x^2)^{1/3}}{x^4} \, dx = \int \frac{(1 - x^2)^{1/3}}{x^{11/3}} \, dx \]

### Step 3: Substitute
Next, we will use the substitution \( t = \frac{1}{x^2} \). Then, we have:

\[ dx = -\frac{1}{2} x^{-3} dt \]

Substituting \( x = \frac{1}{\sqrt{t}} \) gives us:

\[ dx = -\frac{1}{2} \left(\frac{1}{\sqrt{t}}\right)^{-3} dt = -\frac{1}{2} t^{3/2} dt \]

### Step 4: Change the variable in the integral
Now substituting \( x \) and \( dx \) into the integral:

\[ \int \frac{(1 - \frac{1}{t})^{1/3}}{(\frac{1}{\sqrt{t}})^{11/3}} \left(-\frac{1}{2} t^{3/2}\right) dt \]

This simplifies to:

\[ -\frac{1}{2} \int (1 - \frac{1}{t})^{1/3} t^{11/3} t^{3/2} dt \]

### Step 5: Simplify further
Now we can rewrite the integral:

\[ -\frac{1}{2} \int (1 - \frac{1}{t})^{1/3} t^{11/3 + 3/2} dt \]

### Step 6: Integrate
Now we can integrate the expression. The integral can be computed using standard integration techniques, such as integration by parts or substitution.

### Step 7: Back substitute
After integrating, we will substitute back \( t = \frac{1}{x^2} \) to express the answer in terms of \( x \).

### Final Answer
The final answer will be in the form of:

\[ -\frac{3}{8} \left(\frac{1}{x^2} - 1\right)^{4/3} + C \]

where \( C \) is the constant of integration.

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