Problems on Techniques of Integration

By completing the square, we have

\begin{displaymath}ax^2 + b x + c = a \left(\left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a^2} \right)\cdot\end{displaymath}

So

\begin{displaymath}\int \frac{1}{ax^2 + b x + c}dx = \frac{1}{a} \int \frac{1}{\displaystyle \left(x + \frac{b}{2a}\right)^2 + \omega^2}dx\cdot\end{displaymath}

where

\begin{displaymath}\omega = \frac{\sqrt{4ac - b^2}}{2a} \cdot\end{displaymath}

Hence

\begin{displaymath}\int \frac{1}{ax^2 + b x + c}dx = \frac{1}{a \omega} \arctan\left(\frac{x + b/2a}{\omega}\right) + C \end{displaymath}

or

\begin{displaymath}\int \frac{1}{ax^2 + b x + c}dx = \frac{2}{\sqrt{4ac - b^2}} \arctan\left(\frac{2a x + b}{\sqrt{4ac - b^2}}\right) + C .\end{displaymath}

Detailed Answer.


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