Taylor Approximation

1. Mar 29, 2014

Nugso

1. The problem statement, all variables and given/known data

Show that $$∫f'(x)dx/f(x) = ln|(f(x)|+C$$ where f(x) is a differential function.

2. Relevant equations

First order Taylor approximation? $$f(x)=f(a)+f'(a)(x-a)$$

3. The attempt at a solution

Well, I'm not really sure how to approach the question. It's my Numerical Methods homework, so I think I have to do it by using Taylor approximation. By applying the first order Taylor approximation I get:

$$ln|f(x)|=y, ln|f(a)| - ln|(f(x)| = (a-x)f'(x)/f(x)$$

$$ln|(f(a)/f(x)| = (a-x)f'(x)/f(x)$$

I'm kind of stuck here. Am I right in thinking that Taylor approximation is an appropriate way to approach the question?

2. Mar 29, 2014

Ray Vickson

Start again: you are on the wrong track.

3. Mar 29, 2014

Nugso

Hi Ray, thanks for the reply. Would you mind giving me a hint? I really haven't a clue about the right track.

4. Mar 29, 2014

micromass

Why not take the derivative of the right-hand side?

5. Mar 29, 2014

Nugso

I guess my teacher wants us to solve it by the difficult way, i.e using appropriate ways.

6. Mar 29, 2014

Ray Vickson

There is nothing wrong with using the Fundamental Theorem of Calculus to solve the problem. To verify that $\int f(x) \, dx = F(x) + C$, just check that $F'(x) = f(x)$.

That is an absolutely 100% correct way to do the question. In fact, such checking should always be done out of habit, whenever you are faced with a possible formula for the indefinite integral F(x) of an integrand f(x). That is a good way to catch errors.