HMMT 二月 2003 · CALC 赛 · 第 10 题
HMMT February 2003 — CALC Round — Problem 10
题目详情
- Evaluate ∫ 2 ∞ 1 − x dx. 4 −∞ 1 + x 1
解析
- Evaluate ∫ 2 ∞ 1 − x dx. 4 1 + x −∞ Solution: 0 ∫ ∞ 4 Let S = 1 / ( x + 1) dx ; note that the integral converges absolutely. Substituting 0 2 x = 1 /u , so that dx = − 1 /u du , we have ∫ ∫ ∫ 2 ∞ 0 0 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 0 integral from −∞ to ∞ is zero as well. 3