limits..proving they exist???

saint_n

Wot do u have to do to prove that an intergral exists.?? I know how to do it if the integrals bounds are given ( example, [a,b]) but wot if the integral is from x till infinity??

matt grime

In the same wasy as infinite sums, work out the integral from a to b and then let b tend to infinity. Eg
integral of 1/x from a to b is log(b) - log(a), which tends to infinity as b tends to infinity so the integral doesn't exist.
integral of 1/x^2 from a to be is 1/a^2-1/b^2, which tends to 1/a^2 as b tends to infinity so the infinite integral exists.

If you wish to integrate from minus infinity to infinity, you must do the integral from a to b and let a and b tend to infinity independently.

Thus the improper integral of sin(x) over the real line does not exist even though you can choose the interval to be [-a,a] and get an answer of zero (other choices will give different answers hence the integral does not exist)

saint_n

How will you do
[tex]\int\frac{sinx}{x}dx[/tex]
from zero to infinity.
Which can be written as a alternating series
T subscript n =[tex]\mid\int\frac{sinx}{x}dx\mid[/tex] over intervals ([tex](n-1)\pi,n\pi[/tex])
but how do show as n tends to infinity that T(n) tends to 0???
cos i cant integrate it