File:Reciprocal integral.svg

DescriptionReciprocal integral.svg	English: A plot giving some geometric intuition for why the integral of 1/x is ln n, based roughly on an explanation from "Why is the integral of 1/x equal to the natural logarithm of x?". The integral from 1 to 2 is equal to the integral from 2 to 4 and the integral from 4 to 8. Each region is similar to the last except that it has been scaled vertically to 50% its previous height, and scaled horizontally to 200% its previous width. Extending this, the integral from 1 to 2k is k times the integral 1 to 2, just as ln 2k = k ln 2. Source used to generate this chart is shown below.
Date	22 March 2013
Source	Own work   This W3C-unspecified diagram was created with Mathematica.
Author	User:Dcoetzee

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Mathematica source to generate graph (which was then saved as SVG from Mathematica):

Plot[][{{1/x, 1 < x < 2}}],
  Piecewise[{{1/x, 2 < x < 4}}],
  Piecewise[{{1/x, 4 < x < 8}}]},
 {x, 0, 8.5}, PlotRange -> {0, 2.1},
 Filling -> {{2 -> Axis}, {3 -> Axis}, {4 -> Axis}},
 LabelStyle -> {FontSize -> 14},
 AspectRatio -> 0.9,
 Ticks -> {Table[i, {i, 0, 8}], Table[i*0.3, {i, 0, 8}]}]

Axis fonts were converted to Liberation Serif in Inkscape, and the graphic above the plot was added in Inkscape.