Mathematics

Integrate :
$$\displaystyle\int {\dfrac{{dx}}{{x\left( {{x^3} + 1} \right)}}} $$


SOLUTION

Consider the given integral.


$$ I=\displaystyle\int{\dfrac{{{x}^{3}}dx}{{{x}^{4}}\left( {{x}^{3}}+1 \right)}} $$


$$ I=\displaystyle\int{\dfrac{dx}{{{x}^{4}}\left( 1+\dfrac{1}{{{x}^{3}}} \right)}} $$


 


Let $$t=1+\dfrac{1}{{{x}^{3}}}$$


$$ \dfrac{dt}{dx}=0-\dfrac{3}{{{x}^{4}}} $$


$$ -\dfrac{dt}{3}=\dfrac{dx}{{{x}^{4}}} $$


 


Therefore,


$$ I=-\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{t}} $$


$$ I=-\dfrac{1}{3}\ln \left( t \right)+C $$


 


On putting the value of $$t$$, we get


$$I=-\dfrac{1}{3}\ln \left( 1+\dfrac{1}{{{x}^{3}}} \right)+C$$


 


Hence, this is the answer.

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