## anonymous one year ago How do you solve this? Integrate[f'[x]/f[x],{x,a,t}]

1. anonymous

$\int\limits_{a}^{t} \frac{f'(x)}{f(x)} dx$

2. anonymous

Maybe we try to define a function g[x] such that $g'[x] = f'(x) f(x)^{-1}$

3. anonymous

then I can g[t] -g[a]

4. anonymous

by fundamental theorem

5. anonymous

so then I guess I'm asking.. how would I convert $f'(x) f(x)^{-1}$ into g(x) ? Is there a rule or method for it? I thought maybe reverse chain rule.. but I'm not seeing how it would apply.

6. anonymous

oh, irish's post hasnt come through yet. I'll refresh...

7. nincompoop

he had one and now it is gone

8. IrishBoy123

$$\large g(x) = \frac{d}{dx}(ln(f(x)) = \frac{1}{f(x)} \ f'(x)$$ $$G(x) = \int \ g(x) \ dx = \int \ \frac{d}{dx}(ln(f(x)) \ dx\ = ln(f(x))$$

9. anonymous

Did we use the ln function with reverse chain rule here? ok.. I think I get it..

10. anonymous

Thnx irish