Question

\[\int\frac{x^4 + 1}{x^2 + 1} dx\]

Answer 1

\[\int\left( \frac{x^4 + 1}{x^2 + 1} \right)dx\]
\[ = \int\left( \frac{x^4 - 1 + 1 + 1}{x^2 + 1} \right)dx\]
\[ = \int\left[ \frac{\left( x^4 - 1 \right)}{x^2 + 1} + \frac{2}{x^2 + 1} \right]dx\]
\[ = \int\left[ \frac{\left( x^2 - 1 \right)\left( x^2 + 1 \right)}{\left( x^2 + 1 \right)} + \frac{2}{x^2 + 1} \right]dx\]
\[ = \int\left[ \left( x^2 - 1 \right) + \frac{2}{x^2 + 1} \right]dx\]
\[ = \frac{x^3}{3} - x + 2 \tan^{- 1} \left( x \right) + C\]