Question

\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]

Answer 1

\[\int\left( \frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} \right)dx\]
\[ = \int \frac{x^2 \left( 2 x^2 + 7x + 6 \right)}{x\left( x + 2 \right)}dx\]
\[ = \int\frac{x\left[ 2 x^2 + 4x + 3x + 6 \right]}{x + 2}dx\]
\[ = \int\frac{x\left( 2x\left( x + 2 \right) + 3\left( x + 2 \right) \right)}{\left( x + 2 \right)}dx\]
\[ = \int\frac{x\left( 2x + 3 \right)\left( x + 2 \right)}{\left( x + 2 \right)}dx\]
\[ = \int\left( 2 x^2 + 3x \right)dx\]
\[ = 2\int x^2 dx + 3\ intx dx\]
\[ = 2\left[ \frac{x^3}{3} \right] + 3\left[ \frac{x^2}{2} \right] + C\]
\[ = \frac{2}{3} x^3 + \frac{3}{2} x^2 + C\]