MHB Integral involving the derivative

[Yankel]

Hello,

I am trying to calculate the following integral:

\[\int \frac{f'(x)}{f^{2}(x)}dx\]

I suspect is has something to do with the rule of f'(x)/f(x), with the ln, but there must be more to it than that.

can you assist please ? Thank you !

 

[MarkFL]

I would rewrite the given integral as:

$$\int f^{-2}\,df$$

Now just use the power rule. :)

 

[Yankel]

Can you explain your solution please ? How did you go from the derivative and dx to f only with df ?

 

[MarkFL]

$$\int \frac{f'(x)}{f^2(x)}\,dx=\int f^{-2}\d{f}{x}\,dx=\int f^{-2}\,df$$

 

[Yankel]

Right. Simple...

Could you get the same result from the substitution rule ?

 

[MarkFL]

Right. Simple...

Could you get the same result from the substitution rule ?

Yes, I suppose we could write:

$$u=f(x)\implies du=f'(x)\,dx$$

And now we have:

$$\int u^{-2}\,du$$

That seems a little roundabout to me, but perfectly legitimate. :D