HMMT 二月 2009 · CALC 赛 · 第 3 题

英文解析

3. [ 4 ] Compute e where A is defined as
   ∫
   4 / 3
   2 x + x + 12
   dx.
   3 2
   x + x + x + 1
   3 / 4
   Answer:16
   1 x 9
   Solution: We can use partial fractions to decompose the integrand to + , and then integratex +1 x +12
   the addends separately by substituting u = x + 1 for the former and u = x + 1 for latter, to obtain 2
   ∣ √ ∣
   4 / 3
   4 / 3
   1 16
   2 A
   ∣ ∣2
   ln( x + 1) + ln( x + 1) = ln(( x + 1) x + 1) = ln . Thus e = 16 / 9.
   2 9
   3 / 4 3 / 4
   Alternate solution: Substituting u = 1 /x , we find
   ∫ ∫
   3 / 4 4 / 3
   2 3
   2 u + u + u 1 2 /u + 1 + u
   A = ( − ) du = du
   2 3 2 2 3
   1 + u + u + u u 1 + u + u + u
   4 / 3 3 / 4
   Adding this to the original integral, we find
   ∫ ∫
   4 / 3 4 / 3
   2 /u + 2 + 2 u + 2 u 22
   2 A = du = du
   2 3
   1 + u + u + u u
   3 / 4 3 / 4
   16 16
   Thus A = ln and e = .A
   9 9
   ′ ′1