Mathematics

Simplify: $$\displaystyle\int \frac{x^{2}}{x^{3}+1}dx$$


ANSWER

$$\displaystyle \frac{1}{3}\log \left ( 1+x^{3} \right )$$


SOLUTION
Let $$ \displaystyle I=\int  \frac { x^{ 2 } }{ x^{ 3 }+1 } dx$$

Put $$ \displaystyle t=x^{ 3 }+1\Rightarrow dt=\frac { { x }^{ 2 } }{ 3 } dx$$

Therefore 
$$ \displaystyle I=\frac { 1 }{ 3 } \int { \frac { dt }{ t }  } =\frac { 1 }{ 3 } \log { t } =\frac { 1 }{ 3 } \log  \left( 1+x^{ 3 } \right) $$
Hence, option 'A' is correct.
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Single Correct Medium Published on 17th 09, 2020
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