HMMT 二月 2009 · CALC 赛 · 第 8 题
HMMT February 2009 — CALC Round — Problem 8
题目详情
- [ 7 ] Compute √ ∫ 3 ( ) 2 2 2 x +1 2 x +1 2 x x + ln x dx. 1 2
解析
- [ 7 ] Compute √ ∫ 3 ( ) 2 2 2 x +1 2 x +1 2 x x + ln x dx. 1 Answer: 13 ln( x ) Solution: Using the fact that x = e , we evaluate the integral as follows: ∫ ∫ ( ) 2 2 2 x +1 2 2 2 x +1 2 x 2 x +1 2 x +1 2 x + ln x dx = x + x ln( x ) dx ∫ 2 ln( x )(2 x +1) 2 = e (1 + ln( x )) dx ∫ 2 2 x ln( x ) 2 = xe (1 + ln( x )) dx 2 2 2 Noticing that the derivative of x ln( x ) is 2 x (1 + ln( x )), it follows that the integral evaluates to 1 2 2 1 2 x ln( x ) 2 x e = x . 2 2 √ Evaluating this from 1 to 3 we obtain the answer. 2