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Limits you have to know: You are allowed to use the following results without proof: when x , x k e x 0 for any real number k .
when x 0 , x k ln( x) 0 for k 0.
Improper integrals
The integral
b
a
f ( x )dx is said IMPROPER if
a) the interval of integration is infinite, or
b) f ( x) is not defined at one or both of the end points x a and x b.
Method To work out if an improper integral has a value or not (exists or not) 1) Replace "" or "a", the value where f is not defined, by a letter. "N" for example. 2) Integrate to find an expression in terms of "N". 3) Work out the limit of this expression when "N" tends to "" or "a". 4) If the limit exists then the improper integral has a value. If the limit is "", the improper integral does not exist.
1 dxis an improper integral. 1 x2 N 1 N dx Arc tan( x) 0 Arctan(N) Arctan(0) Let's work out 0 1 x2 Example:
0
Arctan(0) = 0 and when N , Arctan(N) conclusion :
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