If f(x)=x+22x+3. then ∫f(x)x21/2dxis equal to 12g1+2f(x)1−2f(x)−23h3f(x)+23f(x)−2 +C where

# If then $\int {\left(\frac{f\left(x\right)}{{x}^{2}}\right)}^{1/2}\mathrm{d}x$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

1. A

$g\left(x\right)={\mathrm{tan}}^{-1}x,h\left(x\right)=\mathrm{log}|x|$

2. B

$g\left(x\right)=\mathrm{log}|x|,h\left(x\right)={\mathrm{tan}}^{-1}x$

3. C

$g\left(x\right)=h\left(x\right)={\mathrm{tan}}^{-1}x$

4. D

$g\left(x\right)=\mathrm{log}|x|,h\left(x\right)=\mathrm{log}|x|$

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### Solution:

Putting ${y}^{2}=f\left(x\right)=\frac{x+2}{2x+3}$, we have

and

So $\begin{array}{l}=-\int y\cdot \frac{2y}{{\left(1-2{y}^{2}\right)}^{2}}\cdot \frac{1-2{y}^{2}}{3{y}^{2}-2}\mathrm{d}y\\ =2\int \frac{{y}^{2}}{\left(2{y}^{2}-1\right)\left(3{y}^{2}-2\right)}\mathrm{d}y\\ =-2\int \left[\frac{1}{2{y}^{2}-1}-\frac{2}{3{y}^{2}-2}\right]\mathrm{d}y\\ =\frac{1}{\sqrt{2}}\mathrm{log}\left|\frac{1+\sqrt{2}y}{1-\sqrt{2}y}\right|-\sqrt{\frac{2}{3}}\mathrm{log}\left|\frac{\sqrt{3}y+\sqrt{2}}{\sqrt{3}y-\sqrt{2}}\right|+C\end{array}$

Thus .

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