Question

2倍正弦加3倍余弦的分子余弦的积分？

Answer 1

2sinx+3cosx = √13[(2/√13)sinx+(3/√13)cosx]
= √13sin(x+t), 其中 t = arctan(3/2)
cosx = cos(x+t-t) = cos(x+t)cost+sin(x+t)sint
I = ∫cosxdx/(2sinx+3cosx)
= (1/√13)∫[cos(x+t)cost+sin(x+t)sint]d(x+t)/sin(x+t)
= (1/√13)∫[costcot(x+t)+sint]d(x+t)
= (1/√13)[cost ln|sin(x+t)| + (x+t)sint] + C, 其中 t = arctan(3/2)

Answer 2

y=2sinx+3cosx z(x)=dy/dx=2cosx-3sinx 令z=0,得x=npi+arctan2/3 n=dz/dx=-2sinx-3cosx 当n>0时y取极大值 当n<0时y取极小值 算起来有点麻烦,本渣要出门抱歉没仔细写