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    Integration: Standard Partition, Integrable over a Range

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    1. Proof. Since 0 x /2, 0 Sin(x) 1.
    So 2 2+Sin(x) 3. Thus,

    (1+x) (1+


    *( /2) (1+ /2)* ( /2)= (2+ )* ( /4)
    as desired.

    2. Solution. By the Stirling formula we have,

    where ~ is used to indicate the ratio of both sides goes to 1 as n tends to infinity.
    By the definition of = and the Stirling formula, we have (as n-> )

    ~ / ...

    Solution Summary

    The integration for standard partition and integrable over a range are determined.