Consider the following expressions:

(1) $\int \frac{f^{\mathrm{\prime }}\left(x\right)}{\left[f\right(x){]}^2}dx=-\frac{1}{f\left(x\right)}+c$

(2) $\int e^{f\left(x\right)}{f}^{\mathrm{\prime }}\left(x\right)dx=e^{f\left(x\right)}+c$

(3) $\int \log \left[f\right(x\left)\right]dx=\frac{1}{f\left(x\right)}+c$

Which of these are correct? 

 Consider first :

     $\frac{f^{\mathrm{\prime }}\left(x\right)}{\left[f\right(x){]}^2}dx$

Let $\mathrm{I}=\int \frac{{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)}{\left[\mathrm{f}\right(\mathrm{x}){]}^2}\mathrm{dx}$

Let $f\left(x\right)=t\Rightarrow f^{\mathrm{\prime }}\left(x\right)dx=dt$

$\Rightarrow I=\int \frac{dt}{t^2}=-\frac{1}{t}+c=-\frac{1}{f\left(x\right)}+c$

So, ( 1) is correct. 

Now consider : 

$\int e^{f\left(x\right)}\cdot f^{\mathrm{\prime }}\left(x\right)dx$

Let $I=\int e^{f\left(x\right)}\cdot f^{\mathrm{\prime }}\left(x\right)dx$

Let $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{t}\Rightarrow {\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{dt}$

$\Rightarrow I=\int e^{t}dt=e^t+c=e^{f\left(x\right)}+c$

So, (2) is also correct. So, (1) and (2) are correct.