HMMT 二月 2009 · CALC 赛 · 第 8 题

英文解析

8. [ 7 ] Compute
   √
   ∫
   3 ( )
   2 2 x +12
   2 x +1 2 xx + ln x dx.
   Answer: 131
   ln( x )
   Solution: Using the fact that x = e , we evaluate the integral as follows:
   ∫ ∫
   ( )
   2 2 x +1 2 22
   2 x +1 2 x 2 x +1 2 x +1 2
   x + ln x dx = x + x ln( x ) dx
   ∫
   ln( x )(2 x +1) 22 = 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 xe = x .
   2 2
   √
   Evaluating this from 1 to 3 we obtain the answer. 2