HMMT 二月 2002 · CALC 赛 · 第 8 题
HMMT February 2002 — CALC Round — Problem 8
题目详情
- Evaluate . 2 (2 x +1) x + x 0 ′
解析
- Evaluate . 2 (2 x +1) x + x 0 √ ∫ 2 x +1 du 2 √ Solution: Let u = x + x . Then du = dx . So the integral becomes 2 , 2 2 (4 x +4 x +1) 2 x + x √ ∫ du − 1 − 1 2 or 2 . This is tan (2 u ), yielding a final answer of tan (2 x + x ) + C for the in- 2 4 u +1 − 1 − 1 π definite integral. The definite integral becomes tan (1) − tan (0) = . 4 ′