Mathematics

Evaluate the following integrals : $$\int \dfrac{x^{2}}{x^{4}+x^{2}+1}dx$$ 


SOLUTION
We have,
$$\begin{array}{l} I=\int { \dfrac { { { x^{ 2 } } } }{ { { x^{ 4 } }+{ x^{ 2 } }+1 } }  } dx \\ =\int { \dfrac { { dx } }{ { { x^{ 2 } }+1+\dfrac { 1 }{ { { x^{ 2 } } } }  } }  }  \\ =\int { \dfrac { { dx } }{ { { { \left( { { x^{ 2 } }+\dfrac { 1 }{ x }  } \right)  }^{ 2 } }-1 } }  }  \\ =\dfrac { 1 }{ 2 } \log  \left( { \dfrac { { x+\dfrac { 1 }{ x } -1 } }{ { x+\dfrac { 1 }{ x } +1 } }  } \right)  \\ =\dfrac { 1 }{ 2 } \log  \left( { \dfrac { { { x^{ 2 } }-x+1 } }{ { { x^{ 2 } }+x+1 } }  } \right) +C \end{array}$$

Hence, this is the answer.
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Subjective Medium Published on 17th 09, 2020
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