Consider where and

MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals

Problem

Consider

\displaystyle I=\int \frac{dx}{ax^{2}+bx+c}

where

a>0

and

b^{2}-4ac>0

. Which of the following gives the correct antiderivative of

I

? (a)

\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)

\dfrac{2}{\sqrt{b^{2}-4ac}}\;\tan^{-1}\left(\dfrac{2ax+b}{\sqrt{b^{2}-4ac}}\right)+C

(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)

\dfrac{1}{\sqrt{b^{2}-4ac}}\;\ln\left|\dfrac{2ax+b+\sqrt{b^{2}-4ac}}{2ax+b-\sqrt{b^{2}-4ac}}\right|+C