Question

\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]

Answer 1

\[\int\left( \frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} \right)dx\]

\[ = \int \frac{x^2 \left( 5 x^2 + 12x + 7 \right)}{x\left( x + 1 \right)}dx\]

 `= ∫ x {( 5 x^2 + 5x + 7x + 7 )}/{( x + 1 )}`

\[ = \int\frac{x\left( 5x\left( x + 1 \right) + 7\left( x + 1 \right) \right)}{\left( x + 1 \right)}dx\]

` = ∫ x{ ( 5x + 7 )( x + 1 )}/{( x + 1 )}dx`

\[ = \int\left( 5 x^2 + 7x \right)dx\]

\[ = \frac{5 x^3}{3} + \frac{7 x^2}{2} + C\]