计算下列不定积分
(1)∫cos3xcos2xdx
(2)∫(x+3)/(x²-5x+6)dx
(3)∫x²lnzdz
(4)∫1/x(3-2lnx)dx
(5)∫e^xsin(x/2)dx

计算下列不定积分
(1)∫cos3xcos2xdx
(2)∫(x+3)/(x²-5x+6)dx
(3)∫x²lnzdz
(4)∫1/x(3-2lnx)dx
(5)∫e^xsin(x/2)dx

(1)∫cos3xcos2xdx=1/2∫(cos5x+cosx)dx =(1/10)sin5x+(1/2)sinx+C (2)由于[(x+3)/(x²-5x+6)]dx=(x+3)/[(x-2)(x-3)] =-5/(x-2)+6/(x-3) ∫(x+3)/(x²-5x+6)dx=∫[-5/(x-2)+6/(x-3)]dx =-5ln|x-2|+6ln|x-3|+C (3)∫x²lnxdx=1/3x³lnx-1/3∫x³•(1/x)dx =(1/3)x³lnx-1/3∫x²dx =(1/3)x³lnx-(1/9)x³+C (4)∫[1/x(3-2lnx)]dx=∫[1/(3-2lnx)](-1/2)d(3-2lnx) =-(1/2)ln|3-2lnx|+C (5)∫e^xsin(x/2)dx=-2∫e^xdcos(x/2) =-2e^xcos(x/2)+2∫e^xcos(x/2)dx =-2e^xcos(x/2)+4∫e^xdsin(x/2) =-2e^xcos(x/2)+4e^xsin(x/2)-4∫e^xsin(x/2)dx 移项得 5∫e^xsin(x/2)dx=2e^x(-cos(x/2)+2sin(x/2))+C 所以 ∫e^xsin(x/2)dx=(2/5)e^x(2sin(x/2)-cosx(x/2))+C