1
$\begingroup$

I can't think of a reason why this may be false.

If I have a linear function $f(x)$ which is strictly increasing (assume $f(x) \neq 0$ and NOT OTHER DISCONTINUTIES ) on a domain $D$, then

$\int_D \dfrac{1}{f(x)} dx = \ln(f(x)) \bigg|_{D}$

No absolute value signs because f(x) is monotonically increasing.

I know it's a rather big claim and I tested it out on $\dfrac{1}{1+x}$ and other similarly simple cases and it see to work.

EDIT: I just noticed $\dfrac{1}{1+x}$ also happens to be monotonically increasing on $(0,\infty)$

EDIT2: Let's tweak the conditions on $D$. Assume x > 0

  • 0
    Sorry, what do you mean by "not other discontinuities"? Is $D$ supposed to be a bounded closed interval?2012-06-02

2 Answers 2

2

$f(x)=\frac{x}{2}>0$ on $[1,2]$ and is strictly increasing, but $\int_1^2\frac{2}{x}\,dx=\ln(4)\ne \ln(2)=\ln(1)-\ln(1/2).$

  • 0
    Sorry, I Fixed it.2012-06-02
0

$\frac{d}{dx}\left( \ln(f(x)) \right) = \frac{f'(x)}{f(x)}.$

When is this equal to $\dfrac{1}{f(x)}$?