# 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+

So,

So,

*( /2) (1+ /2)* ( /2)= (2+ )* ( /4)

as desired.

2. Solution. By the Stirling formula we have,

n!~

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.

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