two improper integrals

Let us consider first the improper integral


The derivativeMathworldPlanetmathPlanetmath I(k) may be formed by differentiating under the integral sign (


Here, the last form has been gotten by the substitution (  kx=t.  But since by the parent entry ( we have


and since  I(0)=0, we can write


Thus we have evaluated the integral I(k):

01-coskxx2𝑑x=πk2. (1)

The formulaMathworldPlanetmathPlanetmath (1) gives


We use here the consequence formula

1-cosx= 2sin2x2

of the double angle formulacos2α=1-2sin2α,  obtaining

π2= 20sin2x2x2𝑑x=0sin2uu2𝑑u,

where the substitution  x2=u  has produced the last form.  Accordingly, we can write as result the formula

0(sinxx)2𝑑x=π2. (2)
Title two improper integrals
Canonical name TwoImproperIntegrals
Date of creation 2013-03-22 18:43:11
Last modified on 2013-03-22 18:43:11
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Example
Classification msc 26A42
Classification msc 26A06
Classification msc 26A03