Question

Simplify: $\int \frac{f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)}{f\left(x\right)g\left(x\right)}dx$

• 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. $\frac{g\left(x\right)}{f\left(x\right)}\log (\frac{g\left(x\right)}{f\left(x\right)})+c$
• D. $\frac{f\left(x\right)}{g\left(x\right)}\log (\frac{g\left(x\right)}{f\left(x\right)})+c$

Answer 1

The correct option is A $\log \frac{g\left(x\right)}{f\left(x\right)}+c$

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

We know that,

$\frac{df\left(x\right)}{dx}=f^{\prime }\left(x\right)\therefore f^{\prime }\left(x\right)dx=df\left(x\right)$

$\therefore I=\int \frac{dg\left(x\right)}{g\left(x\right)}-\int \frac{df\left(x\right)}{f\left(x\right)}=\log \left(g\right(x\left)\right)-\log \left(f\right(x\left)\right)+C\therefore I=\log \left(\frac{g\left(x\right)}{f\left(x\right)}\right)+C$