e3Vx例7.求dxX2 [e3/xd/x =2[e3/xd(3Vx)解：原式=3Vx+C3例8.求sec°xdx.解:原式=「(tan2x +1)2d tan x( (tan * x + 2 tan? x + 1) dtan xx +=tan'x+ tan x +Ctan2目录上页下页返回结束机动
例7. 求 d . 3 x x e x 解: 原式 = e x x 2 d 3 d(3 ) 3 2 3 e x x = e C x = + 3 3 2 例8. 求 sec d . 6 x x 解: 原式 = x xd x 2 2 2 (tan + 1) sec d tan x (tan x 2 tan x 1) dtan x 4 2 = + + x 5 tan 5 1 = x 3 tan 3 2 + + tan x + C
dx例9. 求et+解法1dxd(l+e*)十dx-ldx=r1+ex1+er+= x-ln(l+e")+C解法2-xdxdl+eedx=1+e-x1+e=-ln(l+e-x)+C- In(1 +e-x)= -ln[e-x(ex +1))两法结果一样目录上页下页返回结束机动
例9. 求 . 1 d + x e x 解法1 x e e e x x x d 1 (1 ) + + − = dx = + + − x x e e 1 d(1 ) = x e C x − ln(1+ ) + 解法2 x e e x x d 1 − − + = − − + + = − x x e e 1 d(1 ) e C x = − + + − ln(1 ) − ln(1 + ) = − ln[ ( + 1) ] − x − x x e e e 两法结果一样
例10.求sec xdx解法1dsinxcosxsecxdxdx?- sin? xCOSdsinx1+sinx1-sinx[In|1+ sin x| - In|1- sin x |]+ C1 + sin x-In+C1- sin x目录上页下页返回结束机动
− + + x 1 sin x 1 1 sin 1 2 1 例10. 求 解法1 = x x x d cos cos 2 − = x x 2 1 sin d sin d sin x = ln 1 sin x 2 1 = + − l n 1 − sin x + C C x x + − + = 1 sin 1 sin ln 2 1
secx(secx+tan x)解法2sec xdx =dxsecx+tan xsecx+secxtanxdxsecx+tanxd(sec x + tan x)secx+tanx=lnsec x+tan x+C同样可证csc xdx = In csc x - cot x |+ Cx或+Ccsc xdx = In tan2目录上页下页返回结束机动
+ = sec x tan x 解法 2 sec x + tan x (sec x + tan x) x x x x x x d sec tan sec sec tan 2 + + = d (sec x + tan x) 同样可证 csc xdx = l n csc x − cot x + C 或 C x = + 2 ln tan