# 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

Staff Emeritus
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.