Question

How do you solve this?

Integrate[f'[x]/f[x],{x,a,t}]

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

by fundamental theorem

Answer 5

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.

Answer 6

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

Answer 7

he had one and now it is gone

Answer 8

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

\(G(x) = \int \ g(x) \ dx = \int \ \frac{d}{dx}(ln(f(x)) \ dx\ = ln(f(x))\)

Answer 9

Did we use the ln function with reverse chain rule here?

ok.. I think I get it..

Answer 10

Thnx irish