Question

Solve $\int \frac{x^3+4x^2-7x+5}{x+2}dx$.

Answer 1

$\int \frac{x^3+4x^2-7x+5}{x+2}dx$

$x^2+2x-11$

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$\Rightarrow x+2)x^3+4x^2-7x+5$

$x^3+2x^2$

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$2x^2-7x$ $\Rightarrow \underset{(x+2)(x^2+2x-11)+27}{\underset{\downarrow }{(x^3+4x^2-7x+5)}}$

$2x^2+4x$

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$5-11x$

$-22-11x$

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$27$

$\int (x^2+2x-11+\frac{27}{x+2})dx$

$=\frac{x^3}{3}+x^2-11x+27ln(x+2)+c$.