Evaluate: x^ 2 1 + x^ 3 dx - Vidyadip

Question

$\text{Evaluate: } \int\frac{x^{2}}{1 + x^{3}} \text{dx}$

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Answer

$\text{Let} I \int\frac{x^{2}}{1 + x^{3}} \text{dx}$

$\text{Putting 1} + x^{3} = t$

$\Rightarrow 3x^{2}\text{dx = dt}$

$\text{or x}^{2} dx = \frac{dt}{3} $

$\therefore I = \frac{1}{3} \int \frac{dt}{3} = \frac{1}{3}\log |t| + C $

$= \frac{1}{3} \log | 1 + x^{3}| + C$

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