Find: `int(x^(3))/(sqrt(1-x^(8)))dx`.

Answer

To solve the integral \(\int \frac{x^3}{\sqrt{1 - x^8}} \, dx\), we can follow these steps:

### Step 1: Substitution
Let \( t = x^4 \). Then, we differentiate to find \( dx \):
\[ dt = 4x^3 \, dx \quad \Rightarrow \quad dx = \frac{dt}{4x^3} \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ x^8 = (x^4)^2 = t^2 \]
Thus, we can rewrite the integral as:
\[ \int \frac{x^3}{\sqrt{1 - x^8}} \, dx = \int \frac{x^3}{\sqrt{1 - t^2}} \cdot \frac{dt}{4x^3} \]
This simplifies to:
\[ \int \frac{1}{\sqrt{1 - t^2}} \cdot \frac{dt}{4} \]

### Step 3: Factor Out Constants
We can factor out the constant \(\frac{1}{4}\):
\[ \frac{1}{4} \int \frac{1}{\sqrt{1 - t^2}} \, dt \]

### Step 4: Integrate
The integral \(\int \frac{1}{\sqrt{1 - t^2}} \, dt\) is a standard integral that equals \(\sin^{-1}(t)\):
\[ \frac{1}{4} \sin^{-1}(t) + C \]

### Step 5: Substitute Back
Now, we substitute back \( t = x^4 \):
\[ \frac{1}{4} \sin^{-1}(x^4) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{x^3}{\sqrt{1 - x^8}} \, dx = \frac{1}{4} \sin^{-1}(x^4) + C \]