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Question
\[\int\frac{x^4 + x^2 - 1}{x^2 + 1} \text{ dx}\]
Sum
Solution
\[\int\left( \frac{x^4 + x {}^2 - 1}{x^2 + 1} \right)dx\]
\[ \Rightarrow \int\left( \frac{x^4 + x^2}{x^2 + 1} \right)dx - \int\frac{1}{x^2 + 1}dx\]
\[ \Rightarrow \int\frac{x^2 \left( x^2 + 1 \right)}{\left( x^2 + 1 \right)}dx - \int\frac{1}{x^2 + 1}dx\]
\[ \Rightarrow \int x^2 dx - \int\frac{1}{x^2 + 1}dx\]
\[ \Rightarrow \frac{x^3}{3} - \tan^{- 1} \left( x \right) + C ...........\left( \because \int\frac{1}{x^2 + a^2}dx = \frac{1}{a} \tan^{- 1} \frac{x}{a} + C \right)\]
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