Question

\[\int\frac{x^2 + 5x + 2}{x + 2} dx\]

Answer 1

\[\int\frac{\left( x^2 + 5x + 2 \right)}{\left( x + 2 \right)}dx\]
`=  ∫ x^2 / {x+2}  dx  + 5 ∫   {x   dx} / {x+2 } + 2 ∫  dx/{ x+2}`
\[ = \int\left( \frac{x^2 - 4 + 4}{x + 2} \right)dx + 5\int\left( \frac{x + 2 - 2}{x + 2} \right)dx + 2\int\frac{dx}{x + 2}\]
\[ = \int\frac{\left( x - 2 \right)\left( x + 2 \right)}{\left( x + 2 \right)}dx + \int\frac{4}{x + 2}dx + 5\int\left( 1 - \frac{2}{x + 2} \right)dx + 2\int\frac{dx}{x + 2}\]
\[ = \int\left( x - 2 \right) dx + 4\int\frac{dx}{x + 2} + 5\  ∫ dx - 10\int\frac{dx}{x + 2} + 2\int\frac{dx}{x + 2}\]
\[ = \int\left( x - 2 \right)dx - 4\int\frac{dx}{x + 2} + 5\  ∫  dx\]
\[ = \left( \frac{x^2}{2} - 2x \right) -\text{ 4  ln }\left| x + 2 \right| + 5x + C\]
\[ = \frac{x^2}{2} + 3x - \text{4  ln} \left| x + 2 \right| + C\]