HMMT 二月 2016 · 团队赛 · 第 8 题

英文解析

8. [ 40 ] Compute
   ∫
   2 sin θ + 3 cos θ − 3πd θ.
   13 cos θ − 5
   Proposed by: Carl Lian 0
   3 π 4 3
   Answer: − log
   13 13 2
   We have
   ∫ ∫
   π π/ 2
   2 sin θ + 3 cos θ − 3 2 sin 2 x + 3 cos 2 x − 3
   d θ = 2 d x
   13 cos θ − 5 13 cos 2 x − 5
   0 0
   ∫
   π/ 2
   4 sin x cos x − 6 sin x 2 = 2 d x
   8 cos x − 18 sin x 22
   ∫0
   π/ 2
   sin x (2 cos x − 3 sin x ) = 2 d x
   (2 cos x + 3 sin x )(2 cos x − 3 sin x )
   ∫0
   π/ 2
   sin x = 2 .
   2 cos x + 3 sin x
   To compute the above integral we want to write sin x as a linear combination of the denominator and 0
   its derivative:
   ∫ ∫
   π/ 2 π/ 2 1 − [ − 3(2 cos x + 3 sin x ) + 2(3 cos x − 2 sin x )]
   sin x
   2 = 213
   2 cos x + 3 sin x 2 cos x + 3 sin x
   0 0
   [ ]
   ∫ ∫
   π/ 2 π
   2 − 2 sin x + 3 cos x = − ( − 3) + 2
   13 2 cos x + 3 sin x
   0 0
   [ ]
   2 3 ππ/ 2 = − − + 2 log(3 sin x + 2 cos x ) |
   13 20
   [ ]
   2 3 π 3 = − − + 2 log
   13 2 2
   3 π 4 3 = − log .
   13 13 2