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Today's calculation of Integral 375

Source:

October 6, 2008
calculusintegrationtrigonometryinductioninequalitiescalculus computations

Problem Statement

Prove the following inequality.
1n<∫0π21(1+cos⁡x)n dx<n+5n(n+1) (n=2,3, ⋯ ) \frac {1}{n} < \int_0^{\frac {\pi}{2}} \frac {1}{(1 + \cos x)^n}\ dx < \frac {n + 5}{n(n + 1)}\ (n =2,3,\ \cdots).
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