Question

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

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

Answer 1

The correct option is B $\frac{1}{2}(log\frac{g\left(x\right)}{f\left(x\right)}{)}^2+c$

$\left[\frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f\left(x\right)g\left(x\right)}\right]\left[logg\right(x)-logf(x\left)\right]dx=I$

$I=\int \frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f^2\left(x\right)}\cdot \frac{f\left(x\right)}{g\left(x\right)}\cdot log\frac{g\left(x\right)}{f\left(x\right)}dx$

$I=\int \left[\frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f^2\left(x\right)}\right]\frac{log\frac{g\left(x\right)}{f\left(x\right)}}{g\left(x\right)/f\left(x\right)}dx$

Let $log\frac{g\left(x\right)}{f\left(x\right)}=u$

$\frac{1}{\frac{g\left(x\right)}{f\left(x\right)}}\left[\frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f^2\left(x\right)}\right]dx=du$

$I=\int udu$

$I=\frac{u^2}{2}+c$

$I=\frac{1}{2}{\left(log\frac{g\left(x\right)}{f\left(x\right)}\right)}^2+c$.