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Question
\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]
Sum
Solution
\[\int\left( \frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} \right)dx\]
\[ = \int \frac{x^2 \left( 5 x^2 + 12x + 7 \right)}{x\left( x + 1 \right)}dx\]
`= ∫ 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`
\[ = \int\left( 5 x^2 + 7x \right)dx\]
\[ = \frac{5 x^3}{3} + \frac{7 x^2}{2} + C\]
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