What is the integration of \[ \int {\frac{{{x^2}}}{{1 – {x^4}}}} dx\] ~~~~~[NCERT Exemp. Q. 19,Page 164 ]

Let \[ I = \int {\frac{{{x^2}}}{{1 – {x^4}}}} dx\]

\[ = \int {\frac{{\frac{1}{2}\left( {1 + {x^2}} \right) – \frac{1}{2}\left( {1 – {x^2}} \right)}}{{\left( {1 – {x^2}} \right)\left( {1 + {x^2}} \right)}}} dx\]


\[ = \int {\frac{{\frac{1}{2}\left( {1 + {x^2}} \right)}}{{\left( {1 – {x^2}} \right)\left( {1 + {x^2}} \right)}}} dx – \frac{1}{2}\int {\frac{{\left( {1 – {x^2}} \right)}}{{\left( {1 – {x^2}} \right)\left( {1 + {x^2}} \right)}}} dx\]


\[ = \frac{1}{2}\int {\frac{1}{{1 – {x^2}}}} dx – \frac{1}{2}\int {\frac{1}{{1 + {x^2}}}} dx = \frac{1}{2} \cdot \frac{1}{2}\log \left| {\frac{{1 + x}}{{1 – x}}} \right| + {C_1} – \frac{1}{2}{\tan ^{ – 1}}x + {C_2}\]


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