Question

$\int \frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f\left(x\right)g\left(x\right)}\left[\log \right(g\left(x\right))-\log (f\left(x\right)\left)\right]dx=$

• A. $\log \left(\frac{g\left(x\right)}{f\left(x\right)}\right)+C$
• B. $\frac{1}{2}{\left[\log \left(\frac{g\left(x\right)}{f\left(x\right)}\right)\right]}^2+C$
• C. $\frac{\left(g\right)x}{f\left(x\right)}\log \left(\frac{g\left(x\right)}{f\left(x\right)}\right)+C$
• D. $\log \left[\frac{g\left(x\right)}{f\left(x\right)}\right]-\frac{g\left(x\right)}{f\left(x\right)}+C$

Answer 1

Answer: (B)

$\frac{1}{2}{\left[\log \left(\frac{g\left(x\right)}{f\left(x\right)}\right)\right]}^2+C$

$I=\int \frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f\left(x\right)g\left(x\right)}\left[\log \right(g\left(x\right))-\log (f\left(x\right)\left)\right]dx$
Put $\log \left(g\right(x\left)\right)-\log \left(f\right(x\left)\right)=t$
$\frac{g^{\prime }\left(x\right)}{g\left(x\right)}-\frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=dt$
$\frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f\left(x\right)g\left(x\right)}dx=dt$
$I=\int \frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f\left(x\right)g\left(x\right)}\left[\log \right(g\left(x\right))-\log (f\left(x\right)\left)\right]dx$
$I=\int tdt$
$I=\frac{t^2}{2}$
Subsituting $t,$ we get
$I=\frac{\left[\log \right(g\left(x\right))-\log (f\left(x\right)){]}^2}{2}$
This is same as option $B.$