Integrate the function: x^2 ( 2 + 3 x^3 ) ^3 - Vidyadip

Question

Integrate the function: $\frac{{{x^2}}}{{{{\left( {2 + 3{x^3}} \right)}^3}}}$

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Answer

Let $I = \int {\frac{{{x^2}}}{{{{\left( {2 + 3{x^3}} \right)}^3}}}dx} $$= \frac{1}{9}\int {\frac{{9{x^2}}}{{{{\left( {2 + 3{x^3}} \right)}^3}}}dx} $…(i)
Putting $2 + 3x^3 = t$
$ \Rightarrow 9{x^2} = \frac{{dt}}{{dx}}$
$\Rightarrow 9{x^2}dx = dt$
$\therefore $ From eq. (i), $I = \frac{1}{9}\int\frac{1}{t^3}dt$
$ = \frac{1}{9}\int {{t^{ - 3}}dt} $
$= \frac{1}{9}.\frac{{{t^{ - 2}}}}{{ - 2}} + c$
$= \frac{{ - 1}}{{18{t^2}}} + c$
$= \frac{{ - 1}}{{18{{\left( {2 + 3{x^3}} \right)}^2}}} + c$

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