Integration : Are a function and it's derivative independent?

[jk22]

The question is a bit confused, but it refers to if the following integration is correct :

$$I=\int \frac{1}{1+f'(x)}f'(x)dx$$

$$df=f'(x)dx$$

$$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$

The last equality would come if I suppose $f,f'$ are independent variables.

 

[PeroK]

The question is a bit confused, but it refers to if the following integration is correct :

$$I=\int \frac{1}{1+f'(x)}f'(x)dx$$

$$df=f'(x)dx$$

$$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$

The last equality would come if I suppose $f,f'$ are independent variables.

Try differentiating the result and see whether you get the integrand.

 

[jk22]

I get $$\frac{d}{dx}\frac{f(x)}{1+f'(x)}=\frac{f'(x)}{1+f'(x)}+f(x)\frac{(-1)}{(1+f'(x))^2}f''(x)$$

But if I try : Integrate(1/(1+D(f(x),x)),f) on WolframAlpha I get the above result ?!
Or maybe the partial derivative means something else ?

 

[fresh_42]

You asked initially whether

$$I=\int\frac{f'}{1+f'}dx=?\frac{f}{1+f'}+C$$

and have been told to differentiate. So
$$
\dfrac{d}{dx} I =\dfrac{f'}{1+f'}
$$
and the question is, whether this equals $\dfrac{d}{dx}\left(\dfrac{f}{1+f'}+C\right)$. Does it?