How Do You Prove the Existence of an Improper Integral?

[saint_n]

limits..proving they exist?

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
$$\int\frac{sinx}{x}dx$$
from zero to infinity.
Which can be written as a alternating series
T subscript n =$$\mid\int\frac{sinx}{x}dx\mid$$ over intervals ($$(n-1)\pi,n\pi$$)
but how do show as n tends to infinity that T(n) tends to 0?
cos i can't integrate it