Evaluate: `int((x-x^3)^(1/3))/(x^4)dx`

Explanation

To evaluate the integral

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

we can follow these steps:

### Step 1: Simplify the integrand

First, we can factor 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: Simplify further

Now, we can express the integral as:

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

### Step 3: Use substitution

Let us use the substitution:

\[ t = 1 - x^2 \implies dt = -2x \, dx \implies dx = -\frac{dt}{2x}. \]

From the substitution, we also have:

\[ x^2 = 1 - t \implies x = \sqrt{1 - t}. \]

Thus, we can rewrite \(x^{11/3}\):

\[ x^{11/3} = (1 - t)^{11/6}. \]

### Step 4: Substitute in the integral

Now substituting \(dx\) and \(x^{11/3}\) into the integral gives:

\[ \int \frac{(t)^{1/3}}{(1 - t)^{11/6}} \left(-\frac{dt}{2\sqrt{1 - t}}\right). \]

This simplifies to:

\[ -\frac{1}{2} \int \frac{t^{1/3}}{(1 - t)^{11/6} \sqrt{1 - t}} \, dt. \]

### Step 5: Change the limits and integrate

Now we can integrate:

\[ -\frac{1}{2} \int t^{1/3} (1 - t)^{-8/6} \, dt. \]

Using the integral formula for beta functions or direct integration, we get:

\[ -\frac{1}{2} \cdot \frac{t^{4/3}}{4/3} \cdot (1 - t)^{-2/3} + C. \]

### Step 6: Substitute back

Substituting back \(t = 1 - x^2\):

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

### Final Result

Thus, the final result of the integral is:

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

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