The solution of dydx=x2+4x-9x+2 is

dydx=x2+4x−9x+2⇒ ∫dy=∫(x2+4x−9x+2) dx⇒ ∫dy=∫(x2+4x+4−13x+2) dx⇒ ∫(x+2)2−13x+2 dx⇒∫((x+2)−13x+2) dxy= x22+2x−13log(x+2)+c

The solution of dydx=x2+4x-9x+2 is

dydx=x2+4x−9x+2⇒ ∫dy=∫(x2+4x−9x+2) dx⇒ ∫dy=∫(x2+4x+4−13x+2) dx⇒ ∫(x+2)2−13x+2 dx⇒∫((x+2)−13x+2) dxy= x22+2x−13log(x+2)+c

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The solution of $\frac{dy}{dx} = \frac{x^{2} + 4 x - 9}{x + 2}$ is

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$y = {(x + 2)}^{2} - 13 \log |x + 2| + c$

$y = \frac{x^{2}}{2} + 2 x + 13 \log |x + 2| + c$

$y = {(x + 2)}^{2} - 5 \log |x + 2| + c$

$y = \frac{x^{2}}{2} + 2 x - 13 \log |x + 2| + c$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} = \frac{x^{2} + 4 x - 9}{x + 2} \\ \Rightarrow \int d y = \int (\frac{x^{2} + 4 x - 9}{x + 2}) d x \\ \Rightarrow \int d y = \int (\frac{x^{2} + 4 x + 4 - 13}{x + 2}) d x \\ \Rightarrow \int \frac{{(x + 2)}^{2} - 13}{x + 2} d x \\ \Rightarrow \int ((x + 2) - \frac{13}{x + 2}) d x \\ y = \frac{x^{2}}{2} + 2 x - 13 \log (x + 2) + c \end{array}$