解 ∫(x2 1)sin2xdx= 111(x2 1)dcos2x= (x2 1)cos2x+∫cos2x 2xdx ∫222
11 = (x2 1)cos2x+∫xdsin2x 22
111 = (x2 1)cos2x+xsin2x ∫sin2xdx 222
111 = (x2 1)cos2x+xsin2x+cos2x+C. 224
18. ∫
解 ln3x2x2
11131 = ln3x 3∫ln2xd= ln3x ln2x+3∫dln2x xxxxx∫xln3xdx; 11111dx= ∫ln3xd= ln3x+∫dln3x= ln3x+3∫2ln2xdx xxxxx
131131 = ln3x ln2x+6∫2lnxdx= ln3x ln2x 6∫lnxd xxxxxx
1361 = ln3x ln2x lnx+6∫2dx xxxx
1366 = ln3x ln2x lnx +C. xxxx
19. ∫e
解 ∫e3xdx; 令x=txdx∫t2etdt=3∫t2det
=3t2et 6∫tetdt=3t2et 6∫tdet
=3t2et 6tet+6∫etdt
=3t2et 6tet+6et+C
=3e3x(x2 2x+2)+C.
20. ∫coslnxdx;
解 因为
1 ∫coslnxdx=xcoslnx+∫x sinlnx dx x
1 =xcoslnx+∫sinlnxdx=xcoslnx+xsinlnx ∫x coslnx dx x