Question

lf $f\left(x\right)=\frac{x+2}{2x+3}$ , then $\int (\frac{f\left(x\right)}{x^2}{)}^{1/2}dx$ is
equal to
$\frac{1}{\sqrt{2}}g\left(\frac{1+\sqrt{2f\left(x\right)}}{1-\sqrt{2f\left(x\right)}}\right)-\sqrt{\frac{2}{3}}h\left(\frac{\sqrt{3f\left(x\right)}+\sqrt{2}}{\sqrt{3f\left(x\right)}-\sqrt{2}}\right)+c$
where

• A. $g\left(x\right)={\tan }^{-1}x,h\left(x\right)=\log |x|$
• B. $g\left(x\right)=\log |x|,h\left(x\right)={\tan }^{-1}x$
• C. $g\left(x\right)=h\left(x\right)={\tan }^{-1}x$
• D. $g\left(x\right)=\log |x|,h\left(x\right)=\log |x|$

Answer 1

The correct option is D $g\left(x\right)=\log |x|,h\left(x\right)=\log |x|$

$\int (\frac{f\left(x\right)}{x^2}{)}^{1/2}dx=\frac{1}{\sqrt{2}}g\left(\frac{1+\sqrt{2f\left(x\right)}}{1-\sqrt{2f\left(x\right)}}\right)-\sqrt{\frac{2}{3}}h\left(\frac{\sqrt{3f\left(x\right)}+\sqrt{2}}{\sqrt{3f\left(x\right)}-\sqrt{2}}\right)+C$ .....(1)
where
Given, $f\left(x\right)=\frac{x+2}{2x+3}$
Consider, $\int (\frac{f\left(x\right)}{x^2}{)}^{1/2}dx$
$=\int (\frac{x+2}{2x+3}{)}^{1/2}\frac{dx}{x}$
Put $\frac{x+2}{2x+3}=y^2$
$\Rightarrow x=\frac{3y^2-2}{1-2y^2}$
$\Rightarrow dx=\frac{-2y}{{(1-2y^2)}^2}$
So, $I=-\int \frac{2y^2}{(3y^2-2)(1-2y^2)}dy$
$=\int \frac{2y^2}{(3y^2-2)(2y^2-1)}dy$
Now, we will resolve $\frac{y^2}{(3y^2-2)(2y^2-1)}$ into partial fractions
Put $y^2=t$
So, $\frac{t}{(3t-2)(2t-1)}=\frac{A}{(3t-2)}+\frac{B}{(2t-1)}$
$\Rightarrow t=A(2t-1)+B(3t-2)$
On comparing terms, we get
$2A+3B=1$
$-A-2B=0\Rightarrow A=-2B$
$\Rightarrow B=-1,A=2$
So, $\frac{t}{(3t-2)(2t-1)}=\frac{2}{(3t-2)}-\frac{1}{(2t-1)}$
or,$\frac{y^2}{(3y^2-2)(2y^2-1)}=\frac{2}{(3y^2-2)}-\frac{1}{(2y^2-1)}$
$I=4\int \frac{dy}{(3y^2-2)}-2\int \frac{dy}{(2y^2-1)}$
$I=\frac{4}{3}\int \frac{dy}{(y^2-\frac{2}{3})}-\int \frac{dy}{(y^2-\frac{1}{2})}$
$I=\frac{1}{\sqrt{2}}\ln |\frac{1+\sqrt{2}y}{1-\sqrt{2}y}|-\sqrt{\frac{2}{3}}\ln |\frac{\sqrt{3}y+\sqrt{2}}{\sqrt{3}y-\sqrt{2}}|+C$
$I=\frac{1}{\sqrt{2}}\ln |\frac{1+\sqrt{2f\left(x\right)}}{1-\sqrt{2f\left(x\right)}}|-\sqrt{\frac{2}{3}}\ln |\frac{\sqrt{3f\left(x\right)}+\sqrt{2}}{\sqrt{3f\left(x\right)}-\sqrt{2}}|$
On comparing with (1), we get
$g\left(x\right)=\log |x|,h\left(x\right)=\log |x|$