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Question
Evaluate:\[\int\frac{x^2}{1 + x^3} \text{ dx }\] .
Solution
\[\text{ Let I }= \int \frac{x^2 dx}{1 + x^3}\]
\[\text{ Putting 1} + x^3 = t\]
\[ \Rightarrow 3 x^2 \text{ dx} = dt\]
\[ \Rightarrow x^2 \text{ dx} = \frac{dt}{3}\]
\[ \therefore I = \frac{1}{3}\int \frac{dt}{t}\]
\[ = \frac{1}{3}\text{ ln } \left| t \right| + C\]
\[ = \frac{1}{3}\text{ ln} \left| 1 + x^3 \right| + C \left( \because t = 1 + x^3 \right)\]
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