Question

Assertion :$\int \frac{1}{\sqrt{x^2+2x+10}}dx={\sinh }^{-1}\frac{x+1}{3}+c$
Reason : If $a>0,b^2-4ac<0$, then $\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}{\sinh }^{-1}(\frac{2ax+b}{\sqrt{4ac-b^2}})+k$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

$A=\int \frac{1}{\sqrt{x^2+2x+10}}dx=\int \frac{1}{\sqrt{(x+1{)}^2{+}^2}}dx$

$=\log \mid (x+1)+\sqrt{(x+1{)}^2}+3^2\mid$

$={\sinh }^{-1}\left(\frac{x+1}{3}\right)+c$

$R:\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\log \mid (2ax+b)+\sqrt{(2ax+b{)}^2}+(\sqrt{4ac-b^2{)}^2}\mid$

$=\frac{1}{\sqrt{a}}\sin h^{-1}\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right)$

A, R true and R explains A.