HMMT 二月 2009 · CALC 赛 · 第 10 题

英文解析

10. [ 8 ] Let a and b be real numbers satisfying a > b > 0. Evaluate
    ∫
    2 πdθ.1
    a + b cos( θ )
    Express your answer in terms of a and b .0
    2 π
    √
    Answer:
    2 2
    a − b
    Solution: Using the geometric series formula, we can expand the integral as follows:3
    ( )
    ∫ ∫
    2 π 2 π2
    1 1 b bdθ = 1 + cos( θ ) + cos ( θ ) dθ2
    a + b cos( θ ) a a a
    0 0
    ( ) ( )
    ∫
    ∞nn
    2 πiθ − iθ
    ∑
    1 be + e = dθa a 2
    n =00
    ( )
    ( )
    ∞ n
    2 n
    ∑2
    2 π bn = dθ
    2 2 na a 2
    n =0
    ( )
    2 n
    To evaluate this sum, recall that C = is the n th Catalan number. The generating function 1
    n +1 nnfor the Catalan numbers is
    √
    ∞
    ∑
    1 − 1 − 4 xn
    C x = ,
    2 xnn =0
    )
    ∑ (
    2 nn 1
    √
    and taking the derivative of x times this generating function yields x = . Thus then
    1 − 4 x
    2 π
    √
    integral evaluates to , as desired.
    2 2
    a − b 4