Mathematics

Mathematics

Problem

Consider I=∫dxax2+bx+c\displaystyle I=\int \frac{dx}{ax^{2}+bx+c} where a>0a>0 and b2−4ac>0b^{2}-4ac>0. Which of the following gives the correct antiderivative of II? a) 12b2−4ac  ln⁡∣2ax+b−b2−4ac2ax+b+b2−4ac∣+C\dfrac{1}{2\sqrt{b^{2}-4ac}}\;\ln\left|\dfrac{2ax+b-\sqrt{b^{2}-4ac}}{2ax+b+\sqrt{b^{2}-4ac}}\right|+C b) 2b2−4ac  tan⁡−1(2ax+bb2−4ac)+C\dfrac{2}{\sqrt{b^{2}-4ac}}\;\tan^{-1}\left(\dfrac{2ax+b}{\sqrt{b^{2}-4ac}}\right)+C c) 1b2−4ac  ln⁡∣2ax+b−b2−4ac2ax+b+b2−4ac∣+C\dfrac{1}{\sqrt{b^{2}-4ac}}\;\ln\left|\dfrac{2ax+b-\sqrt{b^{2}-4ac}}{2ax+b+\sqrt{b^{2}-4ac}}\right|+C d) 1b2−4ac  ln⁡∣2ax+b+b2−4ac2ax+b−b2−4ac∣+C\dfrac{1}{\sqrt{b^{2}-4ac}}\;\ln\left|\dfrac{2ax+b+\sqrt{b^{2}-4ac}}{2ax+b-\sqrt{b^{2}-4ac}}\right|+C

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