HMMT 二月 2003 · CALC 赛 · 第 10 题

英文解析

10. Evaluate
    ∫
    ∞2
    1 − xdx.
    1 + x 4 −∞
    Solution: 0
    ∫
    ∞
    Let S = 1 / ( x + 1) dx ; note that the integral converges absolutely. Substituting 4
    x = 1 /u , so that dx = − 1 /u du , we have 20
    ∫ ∫ ∫
    ∞ 0 02
    1 1 du − u
    S = dx = = du
    4 − 4 2 4
    1 + x 1 + u − u u + 1
    0 ∞ ∞
    ∫ ∫
    2 2
    ∞ ∞
    u x = du = dx
    4 4
    1 + u 1 + x
    0 0
    (the manipulations are justified by absolute convergence), from which we see that
    ∫
    ∞
    2 4
    (1 − x ) / (1 + x ) dx = 0. Since the integrand is an even function, it follows that the integral from −∞ to ∞ is zero as well.0 3