HMMT 二月 2016 · 团队赛 · 第 8 题

HMMT February 2016 — Team Round — Problem 8

解析

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