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Question Number 66066 by Kunal12588 last updated on 08/Aug/19

$$Integrate\int \frac{dx}{{ax}^2+bx+c}$$$$y={ax}^2+bx+c$$$$y^{\prime }=\frac{dy}{dx}=2ax+b$$$$d=\sqrt{b^2-4ac}$$$$d^{\prime }=\sqrt{-d}$$$$Case1.d^2<0$$$$I=\frac{2}{d^{\prime }}{tan}^{-1}\frac{y^{\prime }}{d^{\prime }}+C[{tan}^{-1}\alpha =arctan\alpha ]$$$$Case2.d^2=0$$$$I=\frac{-2}{y^{\prime }}+C$$$$Case3.d^2>0$$$$I=\frac{1}{d}\ln \mid \frac{y^{\prime }-d}{y^{\prime }+d}\mid +C[\ln a={\log }_{e}a]$$

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