HMMT 二月 2001 · CALC 赛 · 第 10 题
HMMT February 2001 — CALC Round — Problem 10
题目详情
- Evaluate the definite integral d u .
1+ u 666 − 1
英文原题
Evaluate the definite integral
+1 ∫
− 1
2 u 332 + u 998 +4 u 1664 sin u 691
1+ u 666 d u .
解析
英文解析
- Evaluate the definite integral d u .
1+ u 666 − 1
1664 691
4 u sin u
Solution: The term is odd in u , so its integral is 0. Now make the sub-
1+ u 666 +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