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Integrate the function
$\frac{x^2}{\left(2+3 x^3\right)^3}$
$\frac{x^2}{\left(2+3 x^3\right)^3}$
Solution:
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Verified Answer
Let $2+3 x^3=t \Rightarrow 9 x^2 d x=d t$
$\begin{aligned}
&\therefore \int \frac{x^2}{\left(2+3 x^3\right)^3} d x=\frac{1}{9} \int \frac{d t}{t^3}=\frac{1}{9} \int t^{-3} d t \\
&=-\frac{1}{18 \mathrm{t}^2}+\mathrm{C}=-\frac{1}{18\left(2+3 x^3\right)^2}+\mathrm{C}
\end{aligned}$
$\begin{aligned}
&\therefore \int \frac{x^2}{\left(2+3 x^3\right)^3} d x=\frac{1}{9} \int \frac{d t}{t^3}=\frac{1}{9} \int t^{-3} d t \\
&=-\frac{1}{18 \mathrm{t}^2}+\mathrm{C}=-\frac{1}{18\left(2+3 x^3\right)^2}+\mathrm{C}
\end{aligned}$
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