sint (-sint)dt cost 1+cost t 4sin-cos 2 2 di cost 2 7 2co 2 -2sin2! -joc2aoh=- cost cost =t-In sect+tant+C 1 1,V1-x2 =arccos- .-ln-+ +C X 1 1+V1-x2 arccos -In +C x
8 1 1 1 x dx x x − + 1 sin ( sin ) cos 1 cos t t dt t t = − + 2 2 2 4sin s 1 2 2 cos 2 s 2 t t co dt t t co − = 2 2sin 2 cos t dt t − = cos 1 cos t dt t − = 1 cos dt dt t = − = − + + t t t C ln | sec tan | 2 1 1 1 arccos ln x C x x x − = − + + 2 1 1 1 arccos ln | | x C x x + − = − +
解法3 =1,x 1-t2 1+ -4t (+d Feas - -n-刘--%2 -4t2 -2arctan/-In +t - +c -2arctan, 1-x-In! +-X+C 1+x x 9
9 解法 3 令 1 1 x t x − = + , 2 2 1 1 t x t − = + , 2 2 4 (1 ) t dx dt t − = + 1 1 1 x dx x x − + 2 2 2 2 1 4 1 (1 ) t t t dt t t + − = − + 2 2 2 4 (1 )(1 ) t dt t t − = − + 2 2 2 2 (1 ) (1 ) 2 (1 )(1 ) t t dt t t − − + = − + 2 2 1 1 2 ( ) 1 1 dt t t = − + − 1 2arctan ln 1 t t c t + = − + − 2 1 1 1 2arctan ln 1 | | x x C x x − + − = − + +
例5求 dx xVx2-1 解法l令x=sect,则dk=secttantdt secttantdr 1 =∫df=t+c=arccos-+c 弦25皮-小 1 x2-1=1,x2=12+1,xdx =tdt jwi=j+wajr arctant+c=arctanx2-1+c 10
10 例 5 求 2 1 1 dx x x − 解法 1 令x t = sec ，则dx t tdt = sec tan 2 1 1 sec tan 1 sec tan 1 arccos dx t tdt x x t t dt t c c x = − = = + = + 解法 2 2 2 2 1 1 1 [( 1) 1] 1 dx xdx x x x x = − − + − 令 2 x t − = 1 ， 2 2 x t = +1，xdx tdt = 2 2 2 1 1 1 1 ( 1) 1 dx tdt dt x x t t t = = − + + 2 = + = − + arctan arctan 1 t c x c
可-j小 -dx 2 例6本 e 解令ve-i=t,x=lnr+),=+i 2t di J-rw,=jhe+w +1 11
11 解法 3 2 2 2 1 1 1 1 1 1 dx dx x x x x = − − 2 1 1 1 arccos 1 1 d c x x x = − = + − 例 6 求 1 x x xe dx e − 解 令 1 x e t − = ， 2 x t = + ln( 1)， 2 2 1 t dx dt t = + 1 x x xe dx e − 2 2 2 ln( 1)( 1) 2 1 t t t dx t t + + = + 2 = + 2 ln( 1) t dt