Question

Assertion :If $a>0$ and $b^2-4ac<0$, then the value of the integral $\int \frac{dx}{a{x}^2+bx+c}$ will be of the type $\mu {\tan }^{-1}\frac{x+A}{B}+C$, where $A,B,C,\mu$ are constants. Reason: If $a>0$, $b^2-4ac<0$, then $a{x}^2+bx+C$ can be written as sum of two squares.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

As $a{x}^2+bx+c={ax}^2+bx+{\left(\frac{b}{2a}\right)}^2-{\left(\frac{b}{2a}\right)}^2+c={(ax+\frac{b}{2a})}^2+{\left(\sqrt{c}\right)}^2$
$\therefore \int \frac{dx}{{ax}^2+bx+c}=\int \frac{dx}{{(ax+\frac{b}{2a})}^2+{\left(\sqrt{c}\right)}^2}$
$=\frac{1}{\sqrt{c}}{\tan }^{-1}\frac{ax+\frac{b}{2a}}{\sqrt{c}}=\frac{1}{\sqrt{c}}{\tan }^{-1}\frac{{2a}^{2}x+b}{2a\sqrt{c}}$