Sophomore’s dream

The integral

I:=01xx𝑑x (1)

may be expanded to a rapidly converging series as follows.

Changing the integrand to a power of e and using the power seriesMathworldPlanetmath expansion of the exponential functionDlmfDlmfMathworldPlanetmath gives us

I=01exlnx𝑑x=01n=0(xlnx)nn!dx. (2)

Here the series is uniformly convergent on [0,1] and may be integrated termwise:

I=n=001xn(lnx)nn!𝑑x. (3)

The last equation of the parent entry ( then gives in the case m=n from (3) the result

I=n=0(-1)n(n+1)n+1, (4)


01xxdx= 1-122+133-144+- (5)

Cf. the functionMathworldPlanetmath xx (

Since the series (5) satisfies the conditions of Leibniz’ theorem for alternating seriesMathworldPlanetmath (, one may easily estimate the error made when a partial sum of (5) is used for the exact value of the integral I.  If one for example takes for I the sum of nine first terms, the first omitted term is -11010; thus the error is negative and its absolute valueMathworldPlanetmathPlanetmathPlanetmath less than 10-10.

Title Sophomore’s dream
Canonical name SophomoresDream
Date of creation 2014-07-20 10:46:23
Last modified on 2014-07-20 10:46:23
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Derivation
Classification msc 26A24