Question

if $\int f\left(x\right)dx=f\left(x\right)$, then $\int \left(\frac{f\left(x\right)}{f^{\prime }\left(x\right)}\right)$. dx is equal to

• A. x + c
• B. log f(x) + c
• C. log F(x) + c
• D. $e^{f\left(x\right)}+c$

Answer 1

The correct option is A x + c

$\int f\left(x\right)dx=f\left(x\right)$

Differentiating on both sides

$f\left(x\right)=f^{\prime }\left(x\right)$

$\frac{f\left(x\right)}{f^{\prime }\left(x\right)}=1$

Applying integral on both sides

$\int \frac{f\left(x\right)}{f^{\prime }\left(x\right)}dx=\int dx=x+c$.