Om å integrera éin delt på x

Den naturlege logaritmen

Hans Georg Schaathun

September 2016

\(\int f(x)dx\)\( f(x)\)\( f'(x)\)
\(\vdots\)\(\vdots\)\(\vdots\)
\(\frac14x^4\)\(x^3\)\(3x^2\)
\(\frac13x^3\)\(x^2\)\(2x\)
\(\frac12x^2\)\(x\)\(1\)
\(x\)\(1=x^0\)\(0\)
\( ??? \)\(\frac1x=x^{-1}\)\(-\frac1{x^2}\)
\( -\frac1x \)\(\frac1{x^2}\)\(-2\frac1{x^3}\)
\( -\frac1{2x^2} \)\(\frac1{x^3}\)\(-3\frac1{x^4}\)
\( -\frac1{3x^3} \)\(\frac1{x^4}\)\(-4\frac1{x^5}\)
\(\vdots\)\(\vdots\)\(\vdots\)

$$f(x) = \frac1x$$

$$\ln x $$

\(\ln 1 = 0\)
\(\ln (xy) = \ln x + \ln y\)
\(\ln \frac1x = - \ln x \)
\(\ln \frac xy = \ln x - \ln y \)
\(\ln (x^y) = y\ln x\)