Evaluate : (i) `int(3x^(2))/((1+x^(6)))dx` (ii) `int(x^(3))/((x^(2)+1)^(3))dx` (iii) `(x^(8))/((1-x^(3))^(1//3))dx`

Explanation

To solve the integrals step by step, we will evaluate each of the three given integrals one by one.

### (i) Evaluate \( \int \frac{3x^2}{1+x^6} \, dx \)

1. **Substitution**: Let \( t = x^3 \). Then, \( dt = 3x^2 \, dx \) or \( dx = \frac{dt}{3x^2} \).
2. **Change of Variables**: Substitute \( x^2 = \left( \frac{t}{x} \right)^{2/3} \) and \( 1 + x^6 = 1 + t^2 \).
3. **Rewrite the Integral**:
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