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Integrate the functions x ^2 / 2 + 3 x^3

Q13  Integrate the functions  \frac{x ^2 }{(2+3x^3)^3}

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Given function  \frac{x ^2 }{(2+3x^3)^3},

\int \frac{x ^2 }{(2+3x^3)^3} dx

Assume the 2+3x^3 =t

 \therefore 9x^2dx=dt

\implies \int\frac{x^2}{(2+3x^2)}dx = \frac{1}{9}\int \frac{dt}{t^3}

= \frac{1}{9}\left ( \frac{t^{-2}}{-2} \right ) +C

= \frac{-1}{18}\left ( \frac{1}{t^2} \right )+C

= \frac{-1}{18(2+3x^3)^2}+C  , where C is any constant value.

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