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HMMT 二月 2001 · CALC 赛 · 第 10 题

HMMT February 2001 — CALC Round — Problem 10

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Evaluate the definite integral d u . 666 1+ u − 1
解析
  1. Evaluate the definite integral d u . 666 1+ u − 1 1664 691 4 u sin u Solution: The term is odd in u , so its integral is 0. Now make the sub- 666 1+ u +1 +1 ∫ ∫ 332 998 2 1 2 u + u 1 2+ v 1 / 333 − 332 / 333 stitution u = v ⇒ d u = v d v to find that d u = d v = 666 2 333 1+ u 333 1+ v − 1 − 1 ( ) +1 1 1 ∫ ( ) ∫ ( ) ∫ ( ) 1 1 2 1 2 1 2 2 π − 1 1 + d v = 1 + d v = 1 + d v = (1 + tan 1) = 1 + . 2 2 2 333 1+ v 333 1+ v 333 1+ v 333 333 4 − 1 0 0