Question

If$\int \frac{1}{\sqrt{x^2+x+1}}dx=a{\sinh }^{-1}(bx+c)+d$, then descending order of a, b, c is

• A. a, b, c
• B. b, c, a
• C. b, a, c
• D. c, a, b

Answer 1

The correct option is C b, a, c

$\int \frac{1}{\sqrt{x^2+x+1}}dx=\int \frac{1}{\sqrt{x^2+x+\frac{1}{4}+\frac{3}{4}}}dx$
$=\int \frac{1}{\sqrt{(x+\frac{1}{2}{)}^2+(\frac{\sqrt{3}}{2}{)}^2}}$
$=\log |(x+\frac{1}{2})+\sqrt{x^2+x+1}|+c$
$={\sinh }^{-1}\left(\frac{2x+1}{\sqrt{4-1}}\right)+c$
$={\sinh }^{-1}(\frac{2}{\sqrt{3}}x+\frac{1}{\sqrt{3}})+c$

Comparing with the given equation we get

$a=1,b=\frac{2}{\sqrt{3}},c=\frac{1}{\sqrt{3}}$
Descending order of a,b,c is
b, a, c