Taylor Approximation: Show ∫f'(x)dx/f(x)=ln|f(x)|+C

[Nugso]

Homework Statement

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

Homework Equations

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

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?

 

[Ray Vickson]

Homework Statement

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

Homework Equations

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

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?

Start again: you are on the wrong track.

 

[Nugso]

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

 

[micromass]

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

 

[Nugso]

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

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

 

[Ray Vickson]

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

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.