`int(xsin^(-1)x^(2))/(sqrt(1-x^(4)))dx`

Explanation

To solve the integral \( \int \frac{x \sin^{-1}(x^2)}{\sqrt{1 - x^4}} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \sin^{-1}(x^2) \). Then, we need to find \( dt \) in terms of \( dx \).

### Step 2: Differentiate \( t \)
The derivative of \( t \) is:
\[ dt = \frac{1}{\sqrt{1 - (x^2)^2}} \cdot 2x \, dx = \frac{2x}{\sqrt{1 - x^4}} \, dx \]
From this, we can express \( x \, dx \) in terms of \( dt \):
\[ x \, dx = \frac{dt \cdot \sqrt{1 - x^4}}{2} \]

### Step 3: Substitute in the Integral
Now we can rewrite the integral:
\[ \int \frac{x \sin^{-1}(x^2)}{\sqrt{1 - x^4}} \, dx = \int \sin^{-1}(x^2) \cdot \frac{dt}{2} \]

### Step 4: Simplify the Integral
This simplifies to:
\[ \frac{1}{2} \int t \, dt \]

### Step 5: Integrate
The integral of \( t \) is:
\[ \frac{1}{2} \cdot \frac{t^2}{2} + C = \frac{t^2}{4} + C \]

### Step 6: Substitute Back
Now substitute back \( t = \sin^{-1}(x^2) \):
\[ \frac{1}{4} \left( \sin^{-1}(x^2) \right)^2 + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{x \sin^{-1}(x^2)}{\sqrt{1 - x^4}} \, dx = \frac{1}{4} \left( \sin^{-1}(x^2) \right)^2 + C \]

---