Mathematics

Solve$$\displaystyle\int {\dfrac{{{x^5}}}{{{x^2} + 1}}} dx  $$


ANSWER

$$\dfrac{{{x^4}}}{4} - \dfrac{{{x^2}}}{2} + \dfrac{1}{2}\log ({x^2} + 1) + c$$


SOLUTION

We have,


$$I=\displaystyle\int{\dfrac{{{x}^{5}}}{{{x}^{2}}+1}dx}$$


$$ I=\displaystyle\int{\left( {{x}^{3}}-x+\dfrac{x}{{{x}^{2}}+1} \right)dx} $$


$$ I=\displaystyle\int{{{x}^{3}}dx-\displaystyle\int{x}dx+\dfrac{1}{2}\displaystyle\int{\dfrac{2x}{{{x}^{2}}+1}}dx} $$


$$ I=\dfrac{{{x}^{4}}}{4}-\dfrac{{{x}^{2}}}{2}+\dfrac{1}{2}{{\log }_{e}}\left( {{x}^{2}}+1 \right)+C $$


 


Hence, this is the answer.

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Single Correct Medium Published on 17th 09, 2020
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